抽象代数学习笔记

发表于 2022-02-27 更新于 2022-06-12 分类于笔记阅读次数：

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Preliminaries

equivalence relation

Definition. A binary relation on a set \(A\) is a set \(R \subseteq A \times A\). We denote \((a, b) \in R\) by \(a \sim b\).

Definition. A binary relation \(R\) is an Equivalence relation iff it is

reflexive: \(a \sim a\).

symmetric: \(a \sim b \implies b \sim a\).

transitive: \(a \sim b \land b \sim c \implies a \sim c\).

Definition. For a equivalence relation on \(A\), the equivalence class of \(a \in A\) is defined to be \([a] = \{x \in A \Colon x \sim a\}\) (or denoted by \(\bar{a}\)).

Definition. If \(C\) is an equivalence class, any element \(c \in C\) is called a representative of \(C\).

Definition. A partition of a set \(A\) is a collection \(P = \{A_i | i \in I\}\) of nonempty subsets of \(A\) s.t.

\(A = \cup_{i \in I} A_i\)

\(A_i \cap A_j = \emptyset\) for all \(i, j \in I\) with \(i \ne j\).

If \(P\) is a partition of \(A\) we also say that \(P\) partitions \(A\).

The collection of all equivalence classes is a partition.

residue class

Let \(n\) be a fixed positive integer, \(R\) is a equivalence relation on \(\bbZ\) and \(a \sim b\) iff \(n | (b - a)\).

Definition. \(a\) is congruent to \(b\) mod \(n\) if \(a \sim b\). We write it as \(a \equiv b \pmod{n}\).

Definition. A equivalence class \([a]\) is called a residue class of \(a\) mod \(n\).

Definition. The collection of all equivalence classes is denoted by \(\bbZ_n\) (or \(\bbZ / n \bbZ\)) and called the integers mod \(n\).

Definition. \([a] + [b] = [a + b]\), \([a] \times [b] = [ab]\). They are modular arithmetic.

Definition. \(\bbZ_n^{\times} = \{[a] \in \bbZ_n \Colon (a, n) = 1\}\). It consist the residue classes which have a multiplicative inverse.

Groups

definitions of group

Definition. A binary operation \(\star\) on a set \(G\) is a function \(\star \Colon G \times G \to G\). We denote \(\star(a, b)\) by \(a \star b\).

Definition. A group is a ordered pair \((G, \star)\) where \(G\) is a set and \(\star\) is a binary operation on \(G\) satisfying

\(\star\) is associative.

\(\exists e \in G\) s.t. \(a \star e = e \star a = a\) for all \(a \in G\). \(e\) is called an identity of \(G\) (in fact it is unique).

For each \(a \in G\), \(\exists b \in G\) s.t. \(a \star b = b \star a = e\). \(b\) is usually denoted by \(a^{-1}\) and called an inverse of \(a\) (in fact it is unique).

Less formally, we also say \(G\) is a group under \(\star\) if \((G, \star)\) is a group.

Example. \(\bbZ_n\) is a group under \(+\). \(\bbZ_n^{\times}\) is a group under \(\times\).

Since it is tiresome to keep writing the operation \(\star\), by default, the operation is \(\cdot\) and \(a \cdot b\) is written as \(a b\). And we simply say that \(G\) is a group and we denote the identity of \(G\) by \(1\).

Definition. A group \(G\) is called abelian (or commutative) iff \(a b = b a\) for all \(a, b \in G\).

Definition. For a group \(G\), the order of \(x \in G\) is the smallest positive integer \(n\) s.t. \(x^n = 1\). This integer is denoted by \(|x|\). If such a integer doesn't exist, the order of \(x\) is defined to be infinity.

generators and relations

Let \(G\) be a group.

Definition. A subset \(S \in G\) is a set of generators iff every element \(g \in G\) can be written as a finite product of elements of \(S\) and their inverses. We also say \(S\) generates \(G\) and denote it by \(G = \Inner{S}\).

Definition. If \(G = \Inner{S}\), A relation in \(G\) is an equation in the elements from \(S \cup \{1\}\).

Definition. If there is some collection of relations in \(G = \Inner{S}\), say \(\{\List{m}{R}\}\), s.t. any relation in \(G\) can be deduced from them, then we call these generators and relations a presentation of \(G\) and write \[ G = \Inner{S | \List{m}{R}}. \]

Sometimes \(S\) can be a multiset.

homomorphisms

Definition. If \(G\), \(H\) are two groups under two binary operations, a homomorphism is a map \(\varphi \Colon G \to H\) s.t. \[ \forall x, y \in G \Colon \varphi(x y) = \varphi(x) \varphi(y). \]

\(\varphi(g^n) = \varphi(g)^n\) for all \(n \in \bbZ\) (even negative or zero).

Definition. Let \(\varphi \Colon G \to H\) be a homomorphism, the fiber of \(h \in H\) under \(\varphi\) is defined by \[ \varphi^{-1}(h) := \{g \in G \Colon \varphi(g) = h\}. \]

Definition. Kernel of a homomorphism \(\varphi\) is the fiber \(\varphi^{-1}(1)\), denoted by \(\ker\varphi\).

Both kernel and image of a homomorphism are group (subgroup of \(G\), \(H\), repectively).

Every fiber is a coset of the kernel.

isomorphisms

Definition. An isomorphism is a bijective homomorphism. \(G\) and \(H\) are said to be isomorphic if there exists a isomorphism from one to another. (written as \(G \cong H\)).

(necessary condition) If \(\varphi \Colon G \to H\) is an isomorphism, then

\(|G| = |H|\).
\(G\) is abelian iff \(H\) is abelian.
\(|x| = |\varphi(x)|\) for all \(x \in G\).

It is occasionally easy to see two groups are not isomorphic using the necessary condition above.

A map \(\varphi\) is bijective iff it has two-sided inverse \(\varphi^{-1}\) s.t. \(\varphi \varphi^{-1} = \varphi^{-1} \varphi = 1\), the identity map.

Subgroups

definitions of subgroup

Definition. A group \((H, \star)\) is subgroup of another group \((G, \diamond)\) iff \(H \subseteq G\) and \(\star = \diamond|_{H \times H}\). We denote it by \((H, \star) \le (G, \star)\) or simply \(H \le G\). If \(H \le G\) and \(H \ne G\) we say \(H < G\) (proper subgroup).

If \(H \subseteq G\), \(H\) is a subgroup iff

\(H \ne \emptyset\)
\(xy^{-1} \in H\) for all \(x, y \in H\).

conjugate

Definition. The conjugate of \(n\) (\(N\)) by \(g\) is \(g n g^{-1}\) (\(g N g^{-1}\)). Also we say \(n\) (\(N\)) and \(g n g^{-1}\) (\(gNg^{-1}\)) are conjugate.

For fixed \(g \in G\), the conjugate from \(G\) to \(G\) forms a isomorphism.

centralizer, normalizer, stabilizer

Definition.

Centralizer of subset \(A\) in group \(G\) is defined to be \[ C_G(A) = \{g \in G \Colon gag^{-1} = a \;\;\forall a \in A\}. \]

Normalizer of subset \(A\) in group \(G\) is defined to be \[ N_G(A) = \{g \in G \Colon g A g^{-1} = A\}, \]

Stabilizer of \(s\) (\(s \notin G\) is ok) in group \(G\) is defined to be \[ G_s = \{g \in G \Colon g \cdot s = s\}. \]

Center of \(G\) is defined to be \(Z(G) = C_G(G)\).

\(C_G(A) \le G\), \(G_s \le G\).

Definition. Suppose \(N \le G\). \(g \in G\) normalizes \(N\) iff \(g N g^{-1} = N\). A subgroup \(N \le G\) is called normal iff \(g\) normalizes \(N\) for all \(g \in G\) (\(N_G(N) = G\)). A subset \(A \subseteq G\) normalizes \(N\) iff \(a\) normalizes \(N\) for all \(a \in A\).

\(N\) is normal subgroup of \(G\) is denoted by \(N \unlhd G\). We also say \(N\) is normal in \(G\). Notice that \(\unlhd\) is not transitive in general.

\(N \unlhd G\) iff \(N\) is the kernel of some homomorphism from \(G\).
\(N \unlhd G\) iff \(g N g^{-1} \subseteq N\) for all \(g \in G\).
If \(G = \Inner{S}\) and \(N = \Inner{T}\) is finite (why?) subgroup, \(N \unlhd G\) iff \(s t s^{-1} \in N\) for all \(s \in S, t \in T\).
\(N \unlhd G\) implies \(H N \le G\) for all \(H \le G\) (Recall that \(H N \le G\) iff \(H N = N H\)).

lattice of subgroups

cosets

Definition. For any \(g \in G\), The left coset of \(H\) in \(G\) is \[ gH = \{gh \mid h \in H\}, \] and the right coset is \[ Hg = \{hg \mid h \in H\}. \]

We can usually define a binary operation of two cosets like \[ (g_1 H) \cdot (g_2 H) = (g_1 g_2) H. \]

The above operation is well defined iff \(g h g^{-1} \in H\) for all \(g \in G\) and all \(h \in H\).
\(u H = v H\) iff \(v^{-1} u \in H\).
If \(H \le G\), every coset of \(H\) in \(G\) has size \(|H|\).

Definition. \(A B = \{ab \mid a \in A, B \in B\}\).

Suppose \(A\) and \(B\) are finite subgroups of \(G\), then

\(|A B| = \frac{|A| |B|}{|A \cap B|}\).
the number of distinct ways of writing arbitrary \(x \in AB\) in the form \(x = a b\) is \(|A \cap B|\).
\(A B\) is a subgroup iff \(A B = B A\).

Some Groups

dihedral group

For a ordered regular \(n\)-gon (\(n \ge 3\)) let \(s\) be the reflection about the line of symmetry through vertex \(1\) and the origin, and \(r\) be the rotation clockwise about the origin through \(\frac{2\pi}{n}\) radian.

Definition. \(D_{2n} = \Inner{s, r}\) is called a dihedral group, which is the set of symmetries of regular \(n\)-gon.

\[ D_{2n} = \{1, r, \ldots, r^{n-1}, s, sr, \ldots, sr^{n-1}\} = \Inner{S | s^2 = r^n = 1, s r = r s}. \]

symmetric groups

Let \(\Omega_n = \{1, 2, \ldots, n\}\).

Definition. A permutation of \(A\) is a bijection from \(A\) to itself.

Definition. The set of all permutaions of \(\Omega_n\) is the symmetric group of degree \(n\). we denote it by \(S_n\).

Definition. A cycle \((\List{m}{a})\) is a permutaion which maps \(a_i\) to \(a_{i+1}\) for \(1 \le i \le m - 1\) and maps \(a_m\) to \(a_1\).

\[ (\List{m}{a})^{-1} = (a_m, a_{m-1}, \ldots, a_1). \]

Definition. The cycle decomposition of permutation \(\sigma\) is the product of some disjoint cycles which equals \(\sigma\) like \[ \sigma = (a_1, a_2, \ldots, a_{m_1}) (a_{m_1+1}, a_{m_1+2}, \ldots, a_{m_2}) \ldots (a_{m_{k-1}+1}, a_{m_{k-1}+2}, \ldots, a_{m_k}). \]

Definition. A cycle of length \(t\) is called a \(t\)-cycle. And \(2\)-cycle is called transposition.

\[ (\List{m}{a}) = (a_1, a_m) (a_1, a_{m-1}) \ldots (a_1, a_2) \]

Every permutation can be written as a finite product of (disjoint) cycles, and can be written as a finite product of transpositions.

alternating groups

Definition. Let \(x_1\), \(x_2\), \(x_n\) be indepenendent variables and \(\Delta\) be the polynomial \[ \Delta = \prod_{1 \le i < j \le n} (x_i - x_j) \] Let \(S_n\) act on \(\Delta\) defined by \[ \sigma(\Delta) = \prod_{1 \le i < j \le n} (x_{\sigma(i)} - x_{\sigma(j)}) \] Define the sign of \(\sigma\) by \[ \epsilon(\sigma) = \begin{cases} +1 & \sigma(\Delta) = \Delta \\ -1 & \sigma(\Delta) = \Delta \end{cases} \] \(\sigma\) is called even permutation iff \(\epsilon(\sigma) = +1\) and odd permutation iff \(\epsilon(\sigma) = -1\).

The map \(\epsilon \Colon S_n \to \{\pm 1\}\) is a surjective homomorphism.
Transpositions are all odd permutaions.
A \(t\)-cycle is odd iff \(t\) is even.
A permutation is odd iff the number of cycles of even length in its cycle decomposition is odd.

Definition. The alternating group of degree \(n\), denoted by \(A_n\), is the kerel of the homomorphism \(\epsilon\), that is, the set of even permutaions.

matrix groups

Definition. A field is a set \(F\) together with two binary operations \(+\) and \(\cdot\) s.t. both \((F, +)\) and \((F \setminus \{0\}, \cdot)\) are abelian groups and the following distributive law holds \[ a \cdot (b + c) = (a \cdot b) + (a \cdot c). \]

Definition. General linear group of degree \(n\) is a group under matrix multiplication defined by \[ GL_n(F) = \{A \in M_{n \times n}(F) \Colon \det(A) \ne 0\}. \]

cyclic group

Definition. A group \(G\) is cyclic if \(G = \Inner{x}\) for some \(x\).

Every finite cyclic group of order \(n\) is isomorphic to \(\bbZ_n\), every infinite cyclic group is isomorphic to \(\bbZ\) (\(x^k \mapsto k\)).
Every subgroup of cyclic group is cyclic.
The subgroups of \(\bbZ_n\) correspond bijectively with the positive divisors of \(n\).
\(\Inner{a} = \Inner{(a, n)}\) and \(|a| = \frac{n}{(a, n)}\) for all \(a \in \bbZ_n\).

Quotient Groups

definitions of quotient group

Definition. The collection of all fibers under a homomorphism (with kernel \(K\)) is a group, called quotient group: \[ G / K = \{X = \varphi^{-1}(h) \mid h \in H\}. \] The binary operation of the quotient group is \[ \varphi^{-1}(h_1) \cdot \varphi^{-1}(h_2) \mapsto \varphi^{-1}(h_1 h_2). \]

We can consider the fiber as a equivalence class, and denoted a equivalence class (a coset) by \(\bar{g} = g K\).

Suppose \(K\) is a kernel of some homomorphism from \(G\).

Let \(X \in G / K\) be a fiber, then \(X = uK = Ku\) for all \(u \in X\).
The collection of all left (or right) cosets of \(K\) in \(G\) forms a group. And the group is the quotient group \(G / K\).

\(K\) is a kernel of some homomorphism from \(G\) iff \(K \unlhd G\).

Definition. The homomorphism \(\pi \Colon G \to G / N\) defined by \(g \mapsto g N\) is called the natural projection of \(G\) onto \(G / N\). The complete preimage of \(H \le G / N\) is the preimage of \(H\) under \(\pi\) (that is, \(\pi^{-1}(H)\)).

lagrange's theorem

Definition. The number of left (right) cosets of \(H\) in \(G\) is called the index of \(H\) in \(G\), denoted by \(|G : H|\). Notice that \(H\) is not necessarily normal.

(Lagrange's Theorem) If \(G\) is a finite group and \(H \le G\), then \(|H|\) divides \(|G|\), and \(|G : H| = |G| / |H|\).

Corollary.

If \(G\) is a finite group then \(g^{|G|} = 1\) for all \(g \in G\).
If \(|G|\) is a prime, then \(G\) must be cyclic group.
(Cauchy's Theorem) If \(p\) is a prime dividing \(|G|\), then \(G\) has an element of order \(p\).
(Sylow) If \(|G| = p^a m\) where \(p\) is a prime and \(p\) does not divide \(m\), then \(G\) has a subgroup of order \(p^a\).

the isomorphism theorem

(The 1st Isomorphism Theorem) If \(\varphi \Colon G \to H\) is a homomorphism with kernel \(K\), then \(K \unlhd G\) and \(G / K \cong \varphi(G)\).

(The 2nd Isomorphism Theorem) If \(A \le G\), \(B \le G\), and \(A \le N_G(B)\), then

\(AB \le G\).
\(B \unlhd AB\).
\(A \cap B \unlhd A\)
\((AB)/B \cong A / (A \cap B)\).

In particular, if \(H \le G\) and \(N \unlhd G\), then \[ (HN) / N \cong H / (H \cap N) \]

(The 3rd Isomorphism Theorem) If \(H \unlhd G\), \(K \unlhd G\), and \(H \le K\). Then \(K / H \unlhd G / H\) and \[ (G / H) / (K / H) \cong G / K. \]

To prove that, define a map \(\varphi \Colon G / H \to G / K\) by \(g H \mapsto g K\). It's easy to show \(\varphi\) is well-defined and is a homomorphism with kernel \(K / H\). Then we are done from the 1st theorem.

(The 4th Isomorphism Theorem) If \(N \unlhd G\), then there exists a bijection from the collection of subgroups \(A\) of \(G\) which contain \(N\) to the collection of subgroups \(\bar{A} = A / N\) of \(G / N\).

In particular, every subgroup of \(\bar{G} = G / N\) is of the form \(A / N\) for some subgroup \(A\) of \(G\) containing \(N\).

The "bar" notation above is the bijection, which has the following properties:

\(A \le B \iff \bar{A} \le \bar{B}\).
\(A \le B \implies |B : A| = |\bar{B} : \bar{A}|\).
\(\Up{\Inner{A, B}} = \Inner{\bar{A}, \bar{B}}\).
\(\Up{A \cap B} = \bar{A} \cap \bar{B}\).
\(A \unlhd G \iff \bar{A} \unlhd \bar{G}\).

composition series and the hölder program

Definition. A group \(G\) is called simple group iff \(|G| > 1\) and the only normal subgroups of \(G\) are \(\{1\}\) and \(G\).

Definition. In a group \(G\), a sequence of subgroups \[ 1 = N_0 \le N_1 \le N_2 \le \ldots \le N_{k-1} \le N_k = G \] is called a composition series iff \(N_i \unlhd N_{i+1}\) and \(N_i / N_{i+1}\) is simple for all \(0 \le i \le k - 1\). In that case, each \(N_{i+1} / N_i\) is called composition factor of \(G\).

(Jordan-Holder) If \(G\) is a finite group and \(G \ne \{1\}\), then \(G\) must have a composition series. And every two composition series have the same (isomorphic) composition factors (may in different order).

(Feit-Thompson) If \(G\) is simple group of odd order, then \(G \cong \bbZ_p\) for some prime \(p\).

(The proof of this theorem takes 255 pages of hard mathematics.)

Definition. A group \(G\) is solvable iff there is a chain of subgroups \[ 1 = N_0 \unlhd N_1 \le N_2 \unlhd \ldots \unlhd N_{k-1} \unlhd N_k = G \] such that \(N_{i+1} / N_i\) is abelian for all \(0 \le i \le k - 1\).

Every subgroup and quotient group of solvable group is solvable.
If both \(G / N\) and \(N\) are solvable, then \(G\) is solvable.

Group Actions

definitions of group actions

Definition. A group action of a group \(G\) on a set \(A\) is a function from \(G \times A \to A\) (written as \(g \cdot a\)) satisfying:

\(g_1 \cdot (g_2 \cdot a) = (g_1 g_2) \cdot a\).

\(1 \cdot a = a\).

We say the group \(G\) act on \(A\).

Definition. A permutation representation of \(G\) is any homomorphism from \(G\) to \(S_A\). For a group action of \(G\) on \(A\), let \(\sigma_g(a) = g \cdot a\) (\(\sigma_g \in S_A\)), then the permutation representation associated to the group action is the map \(\varphi \Colon G \to S_A\) defined by \[ \varphi(g) = \sigma_g \] We shall say the group action affords or induces the associated permutation representation of \(G\).

Any permutation representative of \(G\) is a homomorphism.
Any homomorphism from \(G\) to \(S_A\) is associated to a unique group action.

We also call the kernel of \(\varphi\) by the kernel of the associated group action, and it's equal to \(\bigcap_{a \in A} G_a\).

Definition. A group action is called faithful iff its kernel is identity (\(\varphi\) is injective).

Let \(G\) be a group acting on a nonempty set \(A\).

The relation on \(A\) defined by \(a \sim b\) iff \(a = g \cdot b\) for some \(g \in G\), is an equivalence relation.
For each \(a \in A\), the number of elements in \(\bar{a}\) is \(|G \colon G_a|\).

Definition. The equivalence class \(\bar{a} = G a\) is called the orbit of \(G\) containing \(a\).

Definition. A group action is called transitive iff there is only one orbit.

groups acting on themselves by left multiplication

We consider the group action of \(G\) on \(G\) by \((g, a) \mapsto ga\), and action of \(G\) on the collection \(\mathcal{A}_H\) of all left cosets of \(H \le G\) by \((g, a H) \mapsto (ga) H\).

Definition. The associative permutation representation afforded by the actoin of \(G\) on itself is called left regular representation.

Let \(\pi_H\) be the associative permutation representation afforded by the actoin of \(G\) on \(\mathcal{A}_H\).

The group action is transitive.
The stabilizer in \(G\) of the point \(1 H \in \mathcal{A}_H\) is \(G_{1 H} = H\).
\(\ker\pi_H = \bigcap_{g \in G} (g H g^{-1})\), and \(\ker\pi_H\) is the largest normal subgroup of \(G\) contained in \(H\) (any other is subgroup of \(\ker\pi_H\)).

(Cayley's Theorem) Every group is isomorphic to a subgroup of some symmetric group. In particular, if \(|G| = n\) then \(G\) is isomorphic to a subgroup of \(S_n\).

To prove that, assume \(H = \{1\}\), then \(\mathcal{A}_H = G\), \(\pi_H\) is a map from \(G\) to \(S_G\), then \(\ker\pi_H = \bigcap (gHg^{-1}) = \{1\}\), then \(G / \ker\pi_H \cong \pi_H(G) \le S_G\).

If \(G\) is a finite group of order \(n\) and \(p\) is the smallest prime dividing \(|G|\), then any subgroup of index \(p\) is normal.

To prove that, suppose \(H\) is a subgroup of \(G\) of index \(p\). Let \(K = \ker\pi_H\) of index \(k\), then \(|G : K| = |G : H| |H : K| = pk\), \(G / K\) is isomorphic to a subgroup of \(S_p\), \(pk = |G / K|\) divides \(p! = |S_p|\), \(k\) must be \(1\), \(H = K\) is normal.

groups acting on themselves by conjugation

We consider the gorup actoin of \(G\) on \(G\) by \((g, a) \mapsto g a g^{-1}\).

Definition. The orbits of the this action is called conjugacy classes of \(G\).

The number of conjugates of a subset \(A\) in \(G\) is \(|G : N_G(A)|\).

In particular, when \(A = \{a\}\), the number of conjugates is \(|G : C_G(a)|\).

To prove that, recall that the number of element in a orbit cantaining \(s\) is \(|G : G_s|\).

(The Class Equation) If \(G\) is a finite group and \(g_1, g_2, \ldots, g_r\) is representatives of the distinct conjugacy classes of \(G\) not cantained in \(Z(G)\). Then \[ |G| = |Z(G)| + \sum_{i=1}^r |G : C_G(g_i)| \]

If \(p\) is a prime and \(P\) is a group of order \(p^\alpha\) (\(\alpha \ge 1\)), then \(Z(P) \ne \{1\}\). Further, we have \(p\) divides \(|Z(P)|\).

To prove that, just apply the class equation.

Every normal subgroup is a union of conjugacy class.

automorphisms

Definition. An isomorphism from \(G\) onto itself is called an automorphism of \(G\). the set of all automorphisms of \(G\) is denoted by \(\Aut(G)\).

\(\Aut(G)\) is a group under map composition.
\(\Aut(G)\) is isomorphic to a subgroup of \(S_G\).

If \(H \unlhd G\), and let \(G\) acts by conjugation on \(H\). Clearly - Conjugation by \(g\) is an automorphism of \(H\) for all \(g \in G\). - The permutation representation afforded by the action is a homomorphism of \(G\) into \(\Aut(H)\) with kernel \(C_G(H)\). - \(G / C_G(H)\) is isomorphic to a subgroup of \(\Aut(H)\).

Definition. Conjugation by \(g\) is called an inner automorphism of \(G\) and the subgroup of \(\Aut(G)\) consisting of all inner automorphisms is denoted by \(\Inn(G)\).

\(\Inn(G) \unlhd \Aut(G)\).

Definition. A subgroup \(H\) of \(G\) is called characteristic in \(G\), denoted \(H \Char G\), iff every automorphism of \(G\) maps \(H\) to itself.

If \(H\) is the unique subgroup of a given order, then \(H\) is characteristic.
\(H \Char G \implies H \unlhd G\).
\(H \Char K \land K \unlhd G \implies H \unlhd G\).
\(H \Char K \land K \Char G \implies H \Char G\).

sylow's theorem

Definition. Let \(p\) be a prime and \(G\) be a group.

A group of order \(p^\alpha\) for some \(\alpha \ge 1\) is called a \(p\)-group. Subgroup of \(G\) which are \(p\)-groups are called \(p\)-subgroups.

If \(|G| = p^\alpha m\) and \(p\) doesn't divide \(m\), then a subgroup of order \(p^\alpha\) is called a Sylow \(p\)-subgroup of \(G\).

The set of Sylow \(p\)-subgroups of \(G\) is denoted by \(Syl_p(G)\) and the number of Sylow \(p\)-subgroups of \(G\) is denote by \(n_p(G) = |Syl_p(G)|\) (or just \(n_p\) when \(G\) is clear).

Sylow's Theorem is as follows.

(Sylow's Theorem) Suppose \(G\) is a group of order \(p^\alpha m\), where \(p\) is a prime not dividing \(m\).

\(Syl_p(G) \ne \emptyset\).
If \(P \in Syl_p(G)\) and \(Q\) is a \(p\)-subgroup of \(G\), then \(Q \le g P g^{-1}\) for some \(g \in G\). In particular, any two Sylow \(p\)-subgroups of \(G\) are conjugate in \(G\).
\(n_p \bmod p = 1\). Further, \(n_p = |G : N_G(P)|\) for all \(P \in Syl_p(G)\). Hence \(n_p\) divides \(m\).

(Corollary) Let \(P \in Syl_p(G)\), the following are equivalent:

\(n_p = 1\).
\(P \unlhd G\).
\(P \Char G\).
If \(X\) is a arbitrary subset of \(G\) s.t. \(|x|\) is a power of \(p\) for all \(x \in X\), then \(\Inner{X}\) is a \(p\)-subgroup of \(G\).

4 ways to show a group is not simple.

Product of Groups

direct product

Definition. The direct product of \(G_1, G_2, \ldots, G_n\) is \[ G_1 \times G_2 \times \ldots \times G_n = \{(g_1, g_2, \ldots, g_n) \mid g_i \in G_i\}. \] It's a group under "dot product".

Let \(G = G_1 \times \ldots \times G_n\).

\(G_i \cong \{(1, 1, \ldots, g_i, \ldots, 1) \mid g_i \in G_i\} = G_i^* \unlhd G\).
Identify \(G_i\) with \(G_i^*\), we have \[ G / G_i = G / G_i^* \cong G_1 \times \ldots \times G_{i-1} \times G_{i+1} \times \ldots \times G_n \]
Define \(\pi_i \Colon G \to G_i\) by \(g \to g_i\). Then \(\pi_i\) is a surjective homomorphism with kernel isomorphic to \(G / G_i^*\).
If \(x \in G_i^*\), \(y \in G_j^*\), and \(i \ne j\), then \(xy = yx\).

fundamental theorem of finitely generated abelian groups

Definition. A group \(G\) (may infinte) is finitely generated iff there exists a finite subset \(A \subseteq G\) s.t. \(G = \Inner{A}\).

Definition. \(\bbZ^r\) is called the free abelian group of rank \(r\).

(Fundamental Theorem of Finitely Generated Abelian Groups) Let \(G\) be a finitely generated abelian group. Then \[ G \cong \bbZ^r \times \bbZ_{n_1} \times \bbZ_{n_2} \times \ldots \times \bbZ_{n_s} \] for unique integers \(r, n_1, n_2, \ldots, n_s\) satisfying:

\(r \ge 0\) and \(n_j \ge 2\) for all \(j\).
\(n_{i+1}\) divides \(n_i\) for all \(1 \le i \le s - 1\).

Definition. In the note above, \(r\) is called the free rank or Betti number of \(G\). \(s\) is called rank of \(G\) in finite case. The intergers \(n_1, n_2, \ldots, n_s\) is called the invariant factors, and the decomposition is called the invariant factor decomposition of \(G\).

Let \(G\) be an abelian group of order \(n > 1\) and \(n = p_1^{a_1} p_2^{a_2} \ldots p_k^{a_k}\). Then \[ G \cong A_1 \times \ldots \times A_k \] where \(|A_i| = p_i^{a_i}\).

Further, we can decompose \(A_i\) by fundamental theorem of finitely generated abelian groups.

recognizing direct products

Definition. Let \(G\) be a roup, \(x, y \in G\) and \(A, B \subseteq G\), then

Commutator of \(x\) and \(y\) is \([x, y] = x^{-1} y^{-1} x y\).

\([A, B]\) is defined to be \(\Inner{[a, b] \mid a \in A, b \in B}\).

\(G' = \Inner{[x, y] \mid x, y \in G}\) is called commutator subgroup of \(G\).

\(x y = y x [x, y]\) (in particular, \(x\) and \(y\) commute iff \([x, y] = 1\)).
\([x, y]^{-1} = [y, x]\).
\([x, y^{-1}] = y [y, x] y^{-1}\) (in particular, \([x, y] = 1 \iff [x, y^{-1}] = 1\)).
\([x^{-1}, y] = x [y, x] x^{-1}\).

Let \(G\) be a group, \(x, y \in G\), and \(H \le G\), then

\(H \unlhd G\) iff \([H, G] \le H\).
\(\sigma([x, y]) = [\sigma(x), \sigma(y)]\) for all \(\sigma \in \Aut(G)\). And so \(G' \Char G\) and \(G / G'\) is abelian.
\(G' \le H\) iff \(H \unlhd G\) and \(G / H\) is abelian.
If \(\varphi \Colon G \to A\) is a homomorphism of \(G\) into an abelian group \(A\), then \(\varphi\) factors through \(G'\).

PROOF

Trivial.
\(H \unlhd G \iff g^{-1} h g \in H \iff h^{-1} g^{-1} h g \in H \iff [h, g] \in H \iff [H, G] \le H\).
\(\sigma([x, y]) = [\sigma(x), \sigma(y)]\) and \(G' \Char G\) is trivial. Let \(x G', y G' \in G / G'\), clearly \([x G', y G'] = [x, y] G' = G'\), done.
\(H \unlhd G \land [x H, y H] = H \implies [x, y] H = H \implies [x, y] \in H\). Conversely if \(G' \le H\), since \(G / G'\) is abelian, every subgroup of \(G / G'\) is abelian, then \(H / G' \unlhd G / G'\) is abelian, then \(G / H \cong (G / G') / (H / G')\) is abelian.

(recognition theorem) Suppose \(H, K \unlhd G\), and \(H \cap K = \{1\}\), then \(H K \cong H \times K\).

semidirect product

Definition. Let \(H, K\) be groups and \(\varphi\) be a homomorphismsfrom \(K\) into \(\Aut(H)\). Denote the left action of \(K\) on \(H\) determined by \(\varphi\). (i.e. \((k_1 k_2) \cdot h = k_1 \cdot (k_2 \cdot h)\)) Let \(G\) be the set of \((h, k)\) with \(h \in H, k \in K\). Define a binary operation on \(G\) by \[ (h_1, k_1) \cdot (h_2, k_2) = (h_1 (k_1 \cdot h_2), k_1 k_2) \]

This operation makes \(G\) into a group of order \(|H| |K|\).

\(H \cong \{(h, 1) \mid h \in H\} \le G\), \(K \cong \{(1, k) \mid k \in K\} \le G\). And the maps \(h \mapsto (h, 1)\) and \(k \mapsto (1, k)\) are isomorphisms.

\(H \unlhd G\).

\(H \cap K = 1\).

\((k \cdot h, 1) = (1, k) (h, 1) (1, k^{-1})\), can be considered as \(k \cdot h = k h k^{-1}\).

\(G\) is called the semidirect product of \(H\) and \(K\) with respect to \(\varphi\) and be denoted by \(G = H \rtimes_{\varphi} K\) (or just \(H \rtimes K\) when \(\varphi\) is clear).

Let \(H\) and \(K\) be groups and \(\varphi \Colon K \to \Aut(H)\) be a homomorphism, then the following are equivalent:

The identity map between \(H \rtimes K\) and \(H \times K\) is a homomorphism (hence isomorphism).
\(\varphi\) is the trivial homomorphism from \(K\) into \(\Aut(H)\) (i.e. \(k \cdot h = h\)).
\(K \unlhd H \rtimes K\).

Suppose \(G\) is a group, \(H \unlhd G\), \(K \le G\), and \(H \cap K = \{1\}\). Let \(\varphi \Colon K \to \Aut(H)\) be the homomorphism defined by mapping \(k\) to the automorphism of left conjugation by \(k\) on \(H\), then \[ H K \cong H \rtimes K \]

Rings

definitions of rings

Definition. A ring \(R\) is a set together with two binary operation \(+\) and \(\times\) satisfying:

\((R, +)\) is an abelian group (the identity is denoted by \(0\)).

\(\times\) is associative.

The distributive laws hold in \(R\): \[ (a + b) \times c = (a \times c) + (b \times c) \;\;\;\land\;\;\; a \times (b + c) = (a \times b) + (a \times c) \]

If \(a \times b = 0\), then it's a trivial ring. We can see that every abelian group forms a ring.

Definition. A ring \(R\) is commutative iff multiplication is commutative.

Definition. A ring \(R\) has an identity iff there exists an element \(1 \in R\) s.t. \(1 \times a = a \times 1 = a\).

Definition. A ring \(R\) with identity \(1 \ne 0\) is a division ring (or skew field) iff every nonzero element \(a\) has a multiplication inverse \(a^{-1}\).

Let \(R\) be a ring.

\(0 \times a = a \times 0 = 0\).
\((-a) \times b = a \times (-b) = - (a \times b)\).
\((-a) \times (-b) = a \times b\).
If \(R\) has identity \(1\), the identity is unique, and \(-a = (-1) \times a\).

Definition. A nonzero element \(a \in R\) is a zero divisor iff there is a nonzero element \(b \in R\) s.t. either \(a b = 0\) or \(b a = 0\).

Definition. Assume \(R\) has an identity \(1 \ne 0\). An element \(u \in R\) is a unit iff there is some \(v \in R\) s.t. \(u v = v u = 1\) (has inverse).

The set of units is denoted by \(R^{\times}\).

Definition. A commutative ring with identity \(1 \ne 0\) is called an integral domain iff it has no zero divisors.

Assume \(a, b, c \in R\), and \(a\) is not a zero divisor. Then \(a b = a c \implies a = 0 \lor b = c\) (Notice that \(a^{-1}\) may not exist!).

In particular, if \(R\) is integral domain, then \(a b = a c \implies a = 0 \lor b = c\) always holds.

Corollary: Finite integral domain is a field.

subrings

Definition. A subring of a ring \(R\) is a subgroup of \(R\) closed under multiplication.

That is, \(S\) is a subring of \(R\) iff \(x - y \in S\) and \(x y \in S\) for all \(x, y \in S\).

Let \(A, B\) be subrings of \(R\), \(I, J\) be ideals of \(R\).

\(A \cap B\) is a subring.
\(A \cap I\) is a ideal.
\(A + I\) is a subring.
\(I + J\) is a ideal.
\(I \cdot J\) is a ideal.

homomorphisms and isomorphisms

Definition. Let \(R\) and \(S\) be rings.

A ring homomorphism is a map \(\varphi \Colon R \to S\) satisfying \(\varphi(a + b) = \varphi(a) + \varphi(b)\) and \(\varphi(a b) = \varphi(a) \varphi(b)\).

Its kernel is defined to be \(\ker\varphi = \varphi^{-1}(0)\).

If \(\varphi\) is bijection, it's called an isomorphism.

If \(\varphi\) is an isomorphism, we say \(R\) is isomorphic to \(S\) (written as \(R \cong S\)).

Let \(\varphi\) be a ring homomorphism from \(R\) to \(S\).

The image of \(\varphi\) is a subring of \(S\).
The kernel of \(\varphi\) is a subring of \(R\).
\(r k \in \ker\varphi\) and \(k r \in \ker\varphi\) for all \(k \in \ker\varphi\) and \(r \in R\).

quotient ring

Definition. A subring \(I \subseteq R\) is a left ideal iff \(r I \subseteq I\) for all \(r \in R\). Similarly, it's a right ideal iff \(I r \subseteq I\) for all \(r \in R\).

It's a ideal iff it's both a left ideal and a right ideal.

Notice that ideal must be subring by definition.

Definition. Let \(R\) be a ring and \(I\) be an ideal of \(R\). Then we can define a (additive) quotient ring by \[ R / I = \{r + I \mid r \in R\}. \] And it's operation is \[ (r + I) + (s + I) \mapsto (r + s) + I, \] and \[ (r + I) \times (s + I) \mapsto (rs) + I. \]

Let \(I\) be a ideal of a ring \(R\), then the map from \(R\) to \(R / I\) defined by \(r \mapsto r + I\) is a surjective ring homomorphism with kernel \(I\).

Like group case, the map is called natural projection. Thus every ideal is the kernel of a ring and vice versa.

the isomorphism theorem for rings

(The 1st Isomorphism Theorem for Rings) Let \(\varphi \Colon R \to S\) be a homomorphism, then \(\ker\varphi\) is an ideal of \(R\), and \(R / \ker\varphi \cong \varphi(R)\).

(The 2nd Isomorphism Theorem for Rings) Let \(A\) be a subring and \(B\) be an ideal of \(R\), then

\(A + B = \{a + b \mid a \in A, b \in B\}\) is a subring of \(R\).
\(A \cap B\) is an ideal of \(A\).
\((A + B) / B \cong A / (A \cap B)\).

(The 3rd Isomorphism Theorem for Rings) Let \(I\) and \(J\) be ideals of \(R\) with \(I \subseteq J\). Then \(J / I\) is an ideal of \(R / I\) and \[ (R / I) / (J / I) \cong R / J \]

(The 4th Isomorphism Theorem for Rings) Let \(I\) be an ideal of \(R\). The correspondence \(A \leftrightarrow A / I\) is an inclusion preserving bijection between the collection of subrings of \(R\) containing \(I\) and the collection of subrings of \(R / I\).

Further, \(A\) (a subring containg \(I\)) is an ideal of \(R\) iff \(A / I\) is an ideal of \(R / I\).

ideals

Definition. Let \(I\) and \(J\) be ideals of \(R\).

The sum of \(I\) and \(J\) is \[ I + J = \{a + b \mid a \in I, b \in J\}. \]

The product of \(I\) and \(J\) is \[ I \cdot J = \{a_1 b_1 + a_2 b_2 + \cdots + a_n b_n \mid a_i \in I, b_i \in J, n \in \bbN\} \]

Definition. Let \(R\) be a ring with identity \(1 \ne 0\), and \(A\) be an arbitrary subset of \(R\).

The ideal generated by \(A\) is the smallest ideal of \(R\) containing \(A\), denoted by \((A) = \cap_{A \subseteq I} I\).

\((r)\) is called a principal ideal for any \(r \in R\).

\((A)\) is called a finitely generated ideal if \(A\) is finite.

Let \(R\) be a ring with identity \(1 \ne 0\). Let \(I\) be an ideal of \(R\).

\(I = R\) iff \(I\) contains a unit (\(R\) can be generated by any single unit).
Assume \(R\) is commutative. Then \(R\) is a field iff its only ideals are \(\{0\}\) and \(R\).

Corollary: If \(R\) is a field, then any nontrivial homomorphism form \(R\) is an injection.

Definition. A ideal \(M\) of \(R\) is called a maximal ideal iff \(M \ne R\) and the only ideals containing \(M\) is \(R\) and \(M\).

Definition. For a commutative ring \(R\) with \(1\), an ideal \(P\) is called a prime ideal iff \(P \ne R\) and \(a, b \in R, a b \in P \implies \{a, b\} \cap P \ne \emptyset\).

Let \(R\) be a commutative ring with \(1\).

For a ring with \(1\), every proper ideal is contained in a maximal ideal.
The ideal \(M\) is a maximal ideal iff \(R / M\) is a field.
The ideal \(P\) is a prime ideal iff \(R / P\) is an integral domain.
Every maximal ideal is prime ideal (the converse is false in general).

rings of fractions

Let \(R\) be a commutative ring, \(D \subseteq R\) be a nonempty subset s.t. \(0 \notin D\) and \(D\) is closed under multiplication.

Then there is a commutative ring \(Q\) with \(1\) s.t.

\(R\) is a subring of \(Q\).
Every element of \(D\) is a unit in \(Q\).
Every element of \(Q\) is of the form \(\frac{r}{d}\) for some \(r \in R\) and \(d \in D\). In particular, \(D = R \setminus \{0\}\) implies \(Q\) is a field.
\(Q\) is the "smallest" ring containing \(R\) in which all elements of \(D\) becomes units.

Definition. \(Q\) is called the ring of fractions of \(D\) with respect to \(R\), denoted by \(D^{-1} R\). In particular, if \(R\) is an integral domain and \(D = R \setminus \{0\}\) then \(Q\) is called the field of fractions of \(R\).

Let \(Q\) be field of fractions of \(R\). Every field containing \(R\) must contain \(Q\) (\(Q\) is the "smallest" field containg \(R\)).

For example, \(\bbQ\) is the smallest field containg \(\bbZ\).

the chinese remainder theorem

Definition. The ideals \(A\) and \(B\) of the ring \(R\) are said to be comaximal iff \(A + B = R\).

Let \(A_1, A_2, \ldots A_k\) be ideals of \(R\). The map \(R \to R / A_1 \times R / A_2 \times \cdots \times R / A_k\) defined by \(r \mapsto (r + A_1, r + A_2, \ldots, r + A_k)\) is a ring homomorphisms with kernel \(A_1 \cap A_2 \cap \cdots \cap A_k\).

If \(A_i\) and \(A_j\) are comaximal for all \(i \ne j\), then this map is surjective and the kernel equal to \(A_1 A_2 \cdots A_k\).

Some Rings

quadratic integer rings

Definition. \(\bbZ[\omega] = \{a + b \omega \mid a, b \in \bbZ\}\).

If \(D\) is squarefree, then \[ \bbZ[\sqrt{D}] \cong \{\begin{pmatrix} a & b \\ Db & a \end{pmatrix}\}. \] The right-hand side is a subring of \(M_2(\bbZ)\).

If \(D = 4k + 1\) is squarefree, then \[ \bbZ[\frac{1+\sqrt{D}}{2}] \cong \{\begin{pmatrix} a & b \\ kb & a + b \end{pmatrix}\}. \] The right-hand side is a subring of \(M_2(\bbZ)\).

polynomial rings

Definition. Let \(R\) be a commutative ring \(R\) with identity. Polynomial ring is defined as follows: \[ R[x] = \{a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \mid a_i \in R, n \in \bbN\}. \]

The formal sum above is called polynomial of degree \(n\).

Let \(R\) be an integral domain and \(0 \ne p(x), q(x) \in R[x]\).

\(\text{degree } p(x) q(x) = \text{degree } p(x) + \text{degree } q(x)\).
The units of \(R[x]\) are just the units of \(R\).
\(R[x]\) is an integral domain.

matrix rings

Definition. Let \(R\) be a arbitrary ring, then \(M_{n \times n} (R)\) is a ring, called matrix ring. Sometimes we denote it by \(M_n(R)\).

group rings

Definition. Let \(R\) be a commutative ring with identity \(1 \ne 0\), and \(G = \{g_1, g_2, \ldots, g_n\}\) be a finite group. Then \(RG\) is the ring of all following formal sums (\(a_i \in R\)) \[ a_1 g_1 + a_2 g_2 + \ldots + a_n g_n \] which is called group ring.

Some Domains

Recall that integral domain is commutative with identiy, and has no zero divisors.

Euclidean Domains

Definition. Let \(R\) be a integral domain.

A function \(N \Colon R \to \bbN\) with \(N(0) = 0\) is called a norm on \(R\).

If \(N(a) > 0\) for all \(a \ne 0\) then it's called a positive norm.

\(R\) is said to be a Euclidean Domain iff there exists a norm \(N\) s.t. for all \(a, b \in R\) with \(b \ne 0\), there exists \(q, r \in R\) with \(a = q b + r\) s.t. \(r = 0\) or \(N(r) < N(b)\).

\(q, r\) is called the quotient and the remainder of the division.

If \(a = u b\) for some unit \(u\), then we say \(a\) and \(b\) are associate.

Definition. Let \(R\) be a commutative ring.

\(a\) is said to be a multiple of \(b\) (or \(b\) divides \(a\)) iff \(\exists x \in R\) s.t. \(a = b x\), written \(b \divide a\).

\(d\) is a greatest common divisor of \(a\) and \(b\) iff \((d \divide a) \land (d \divide b)\) and \((d' \divide a) \land (d' \divide b) \implies d' \divide d\).

Let \(R\) be a commutative ring.

\(b \divide a \iff a \in (b) \iff (a) \subseteq (b)\).
If \((d) = (a, b)\), then \(d\) is a greatest common divisor.
\((a, b) = (a) + (b)\).

Let \(R\) be an integral domain.

If \((d) = (d')\), then \(d'\) and \(d\) are associate. In particular, every two gcds of \(a, b\) are associate.
If \((d) = (a, b)\) then \(d\) can be written as \(ax + by\) for some \(x, y \in R\).

Definition. Euclidean Algorithm for \(a, b\) is:

Let \(r_0 = a, r_1 = b\).

Take division \(r_i = q_{i+2} r_{i+1} + r_{i+2}\) for \(i\),

Until \(r_i = 0\).

\(F[x]\) is a Euclidean Domain with \(N\) being the degree of polynomial, where \(F\) is a field.
\(\bbZ[\sqrt{-1}]\) is a Euclidean Domain with \(N(a + b \sqrt{-1}) = a^2 + b^2\).

Principal Ideal Doamins

Definition. A Principal Ideal Domain is an integral domain in which every ideal is principal.

\(\bbZ[\sqrt{-5}]\) is not a Principal Ideal Domain because \((3, 1 + \sqrt{-5})\) is not principal.

Every Euclidean Domain is a Principal Ideal Domain. And \(I = (d)\) for all \(d\) having minimum norm in \(I\).

In general a Principal Ideal Domain is not a Euclidean Domain. \(\bbZ[\frac{1 + \sqrt{-19}}{2}]\) is an example.

Let \(R\) be a Principal Ideal Domain, and \(a, b \in R\) are nonzero. If \((a, b) = (d)\), then

(Recall) \(d\) is a greatest common divisor of \(a\) and \(b\).
(Recall) \(d\) can be written as \(d = ax + by\).

If \(R\) is a Euclidean Domain, and \(r_n\) is the last nonzero remainder in the Euclidean Algorithm for \(a\) and \(b\), then \((r_n) = (a, b)\).

Every nonzero prime ideal in a Principal Ideal Domain is a maximal ideal.
If \(R\) is a commutative ring and \(R[x]\) is a Principal Ideal Domain, then \(R\) must be a field.

Unique Factorization Domains

Definition. Let \(R\) be an integral domain.

Let \(r \in R\) be nonzero and not a unit, then \(r\) is called reducible in \(R\) iff \(r = a b\) for some \(a, b\) which are not units. Otherwise, it's called irreducible.

The nonzero element \(p \in R\) is called prime in \(R\) iff the ideal \((p)\) is a prime ideal.

If \(R\) is an integral domain, \(p\) is prime \(\implies\) \(p\) is irreducible.
If \(R\) is a Principal Ideal Domain, \(p\) is prime \(\iff\) \(p\) is irreducible.

Definition. A Unique Factorization Domains is an integral domain \(R\) s.t. every nonzero element \(r\) which is not a unit obeys:

\(r\) can be written as \(p_1 p_2 \cdots p_n\) (finite), where all \(p_i\) are irreducible.

The decomposition above is unique up to associates.

Let \(R\) be a Unique Factorization Domain, and \(a, b \in R\). Suppose \(a = u p_1^{f_1} p_2^{f_2} \cdots p_n^{f_n}\), and \(b = v p_1^{g_1} p_2^{g_2} \cdots p_n^{g_n}\).

The greatest common divisor \(d\) alwasy exists, and \(d\) is associated to \(p_1^{h_1} p_2^{h_2} \cdots p_n^{h_n}\) where \(h_i = \min(f_i, g_i)\).

Every Principal Prime Ideal is a Unique Factorization Domain.

(Factorization in the Gaussian Integers)

Fermat's Theorem on sums of squares: the prime \(p\) can be written as \(a^2 + b^2\) iff \(p = 2\) or \(p \bmod 4 = 1\). And \((a, b)\) is unique.

The irreducible elements in \(\bbZ[i]\) are as follows:

\(1 + i\).
prime \(p\), where \(p \bmod 4 = 3\).
\(a \pm bi\), where \(a^2 + b^2\) is a prime.

Polynomial Rings

basic properties

Let \(I\) be an ideal of the ring \(R\), then

\((I) = I[x]\) is an ideal of \(R[x]\).
\(R[x] / I[x] \cong (R / I)[x]\).

In particular, if \(I\) is a prime ideal of \(R\), then \((I) = I[x]\) is a prime ideal of \(R[x]\).

(Recall) \(F[x]\) is a Euclidean Domain when \(F\) is a field.

over Unique Factorization Domain

Definition. A polynomial is monic iff \(a_n = 1\).

(Gauss' Lemma) Let \(R\) be a Unique Factorization Domain, \(F\) be a field of fractions of \(R\), and let \(p(x) \in R[x]\).

\(p(x)\) is reducible in \(F[x]\) \(\implies\) \(p(x)\) is reducible in \(R[x]\).
(Corollary) If the greatest common divisor of the coefficients of \(p(x)\) is \(1\), then \(p(x)\) is irreducible in \(R[x]\) \(\iff\) \(p(x)\) is irreducible in \(F[x]\).

\(R\) is a Unique Factorization Domain \(\iff\) \(R[x]\) is a Unique Factorization Domain.

irreducible criteria

Let \(F\) be a field and \(p(x) \in F[x]\).

\(p(x)\) has a factor of degree one \(\iff\) \(p(x)\) has a root in \(F\).
If the degree of \(p(x)\) is \(2\) or \(3\), \(p(x)\) is reducible \(\iff\) it has a root in \(F\).

Let \(p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 \in \bbZ[x]\).

If \(r / s \in \bbQ\) with \((r, s) = 1\) is a root of \(p(x)\), then \(r \divide a_0\) and \(s \divide a_n\).

Let \(I\) be a proper ideal in integral domain \(R\), and \(p(x)\) be a nonconstant monic polynomial in \(R[x]\). If the image of \(p(x)\) in \((R / I)[x]\) cannot be factored in \((R / I)[x]\), then \(p(x)\) is irreducible in \(R[x]\).

(Eisenstein's Criterion) Let \(P\) be a prime ideal of the integral domain \(R\), and \(f(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_0 \in P[x]\), and \(a_0 \notin P^2\). Then \(f(x)\) is irreducible in \(R[x]\).

(Corollary when \(R = \bbZ\)) Let \(p\) be a prime integer, and \(f(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_0\) s.t. \(p \divide a_i\) but \(p^2\) doesn't divide \(a_0\). Then \(f(x)\) is irreducible in \(\bbZ[x]\) and \(\bbQ[x]\).

over field

The maximal ideals in \(F[x]\) are the ideals \((f(x))\) generated by irreducible polynomials \(f(x)\).

In particular, \(F[x] / (f(x))\) is a field iff \(f(x)\) is irreducible.

TODO: write more.

Module Theory

basic definitions

Definition. Let \(R\) be a ring. A (left) module over \(R\) is a set \(M\) together with

a binary operation \(+\) on \(M\) s.t. \(M\) is an abelian group under \(+\)

an action of \(R\) on \(M\) (ring action) s.t. for arbitrary \(r, s \in R\) and \(m, n \in M\):

\((r + s) \cdot m = r \cdot m + s \cdot m\).

\((rs) \cdot m = r \cdot (s \cdot m)\).

\(r \cdot (m + n) = r \cdot m + r \cdot n\).

If \(R\) has \(1\), every module satisfying \(1 \cdot m = m\) called a unital module.

If \(N \subseteq M\) and \(N\) is a module together with the same binary operation and the same action, then \(N\) is a submodule of \(M\).

Warning! We only consider the unital module.

Every abelian group \(A\) is a module over \(\bbZ\) (called \(\bbZ\)-module).

Let \(R\) be a ring and \(M\) be an module. A subset \(N \subseteq M\) is a submodule of \(M\) iff

\(N \ne \emptyset\).
\(x + r \cdot y \in N\), for \(r \in R\) and \(x, y \in N\).

Definition. Let \(R\) be a commutative ring with \(1\). An \(R\)-algebra is a ring \(A\) with identity together with a ring homomorphism \(f \Colon R \to A\) mapping \(1_R\) to \(1_A\) s.t. \(f(R)\) is a subring of \(A\) contained in \(Z(A)\).

If \(A\) and \(B\) are \(R\)-algebras, an \(R\)-algebra homomorphism is a ring homomorphism \(\varphi \Colon A \to B\) s.t.

\(\varphi(1_A) = 1_B\).

\(\varphi(r a) = r \varphi(a)\).

Also, an isomorphism is a bijective homomorphism.

Let \(A\) be a ring with \(1_A\) and a \(R\)-module satisfying \(r \cdot (a b) = (r \cdot a) b\). Then \(f \Colon R \to A\) defined by \(r \mapsto r \cdot 1_A\) makes \(A\) be an \(R\)-algebra.

\(A\) is an \(R\)-algebra iff

\(A\) is a ring with \(1_A\).
\(A\) is a \(R\)-module.
\(r \cdot (a b) = (r \cdot a) b = a (r \cdot b)\) (they're equal to \(f(r) a b\)).

quotient modules and module homomorphisms

Definiton. Let \(R\) be a ring and \(M, N\) be \(R\)-modules.

A map \(\varphi \Colon M \to N\) is an \(R\)-module homomorphism satisfying

\(\varphi(x + y) = \varphi(x) + \varphi(y)\).

\(\varphi(r \cdot x) = r \cdot \varphi(x)\).

An \(R\)-module homomorphism is an isomorphism iff it's bijective.

\(\Hom_R(M, N)\) is defined to be the set of all \(R\)-module homomorphisms from \(M\) into \(N\).

Let \(\varphi, \psi \in \Hom_R(M, N)\), define \(\varphi + \psi\) by \(m \mapsto \varphi(m) + \psi(m)\).

Let \(\varphi \in \Hom_R(M, N)\), define \(r \cdot \varphi\) by \(m \mapsto r \cdot \varphi(m)\).

If \(M = N = R\), a ring homomorphism may not be \(R\)-module homomorphism, also a \(R\)-module homomorphism may not be ring homomorphism. For example,

if \(R = \bbZ\) then \(x \mapsto 2x\) is a \(R\)-module homomorphism but not a ring homomorphism.
if \(R = F[x]\) then \(f(x) \mapsto f(x^2)\) is a ring homomorphism but not a ring homomorphism.

\(\varphi \in \Hom_R(M, N) \iff \varphi(r x + y) = r \varphi(x) + \varphi(y) \;\;\forall r,x,y\).
\(\varphi + \psi \in \Hom_R(M, N)\), and so \(\Hom_R(M, N)\) is an abelian group under this addition.
If \(R\) is commutative, then \(r \cdot \varphi \in \Hom_R(M, N)\), and so \(\Hom_R(M, N)\) is an \(R\)-module.
If \(\varphi \in \Hom_R(L, M)\) and \(\psi \in \Hom_R(M, N)\), then \(\varphi \circ \psi \in \Hom_R(L, N)\).
With addition above and multiplication as function composition, \(\Hom_R(M, M)\) is a ring with \(1\), and is a \(R\)-algebra if \(R\) is commutative.

Let \(R\) be a a ring, \(M\) be an \(R\)-module, and \(N\) be a submodule of \(M\).

The quotient group \(M / N\) can be made into an \(R\)-module by defining an action: \(r \cdot (x + N) = (r \cdot x) + N\).

The natrual projection \(\pi \Colon M \to M / N\) defined by \(\pi(x) = x + N\) is an \(R\)-module homomorphism with kernel \(N\).

(isomorphism theorems)

\(M / \ker\varphi \cong \varphi(M)\).
\((A + B) / B \cong A / (A \cap B)\).
\((M / A) / (B / A) \cong M / B\).
\(A \leftrightarrow A / N\).

generation of modules, direct sums, and free modules

Definiton. Let \(M\) be an \(R\)-module, \(N_1, N_2, \ldots, N_n\) be submodules of \(M\), and \(A\) be a subset of \(M\).

\(N_1 + N_2 + \cdots + N_n = \{a_1 + a_2 + \cdots + a_n \mid a_i \in N_i\}\).

\(R A = \{r_1 a_1 + r_2 a_2 + \cdots + r_m a_m \mid r_i \in R, a_i \in A, m \in \bbN\}\).

If \(R A = N\) we call \(A\) a set of generators or generating set for \(N\), and we say \(N\) is generated by \(A\).

\(N\) is finitely generated if there is some finite \(A\) s.t. \(N = R A\).

\(N\) is cyclic if there is some \(A = \{a\}\).

Field Theory

basic definition

Definition. The characteristic of a field \(F\) is \(\ch(F) = \min\{p \in \bbN \colon p \cdot 1_F = 0_F\}\), and \(\ch(F) = 0\) if it's an empty set.

Definition. The prime subfiled of a field \(F\) is the subfield generated by \(1_F\).

Definition. For a field \(F\), the subfield generated by a subset \(A\) is defined to be the insection of all subfield containing \(A\).

Definition. If \(F\) is a subfield of \(K\), then \(K\) is said to be an extension field of \(F\), denoted \(K / F\).

Definition. The degree of \(K / F\), denoted \([K : F]\), is the dimension of \(K\) as a vector space over \(F\).

\(\ch(F)\) is either \(0\) or a prime.
If \(\ch(F) = 0\) then the subfield generated by \(1_F\) is \(\bbQ\).
If \(\ch(F) = p \ne 0\) then the subfield generated by \(1_F\) is \(\bbF_p\).

A homomorphism \(\varphi \colon F \to F'\) is either \(0\) or injective, with image either \(\{0\}\) or isomorphic to \(F\).

If \(p(x) \in F[x]\) is irreducible, then there exists a field \(K\) contains an isomorphic copy of \(F\), and \(p(x)\) has a root in \(K\).

In fact we can suppose \(K = F[x] / (p(x))\), and the root is just \(x\) in \(K\).

We can consider \(K\) as a vector space over \(F\), if the degree of \(p(x)\) is \(n\), then the basis of the vector space is \[ \begin{aligned} 1, x, x^2, \ldots, x^{n-1}. \end{aligned} \]

For example, \(x^2 + 1 \in \bbR[x]\) is a irreducible polynomial, and it has root in \(\bbR[x] / (x^2 + 1) \cong \bbC\).

Definition. Let \(K\) be an extension of \(F\) and \(A \subset K\), then the smallest subfield of \(K\) containing \(F \cup A\), denoted \(F(A)\), is called the field generated by \(A\) over \(F\).

If \(K = F(a)\) then \(K\) is said to be a simple extension of \(F\), and \(a\) is called a primitive element for the extension.

\(p(x) \in F[x]\) is irreducible. An extension \(K / F\) contains a root \(a\) of \(p(x)\). Then \(F(a) \cong F[x] / (p(x))\).

That is, \(F[x] / (p(x))\) is the smallest field containing a root.

\(\varphi \colon F \to F'\) is a isomorphism,
\(p(x) \in F[x]\) is irreducible,
\(p'(x) \in F'[x]\) is obtained by applying the map \(\varphi\) to the coefficients of \(p(x)\).
\(a\) is a root of \(p(x)\) in some extension of \(F\).
\(a'\) is a root of \(p'(x)\) in some extension of \(F'\).

Then there is an isomorphism \(\sigma \colon F(a) \to F'(a')\) such that

\(\sigma(a) = a'\).
\(\sigma\) is a extension of \(\varphi\).

algrbraic extension

Definition. \(\alpha \in K\) is said to be algebraic over subfield \(F\) if \(\alpha\) is a root of some \(p(x) \in F[x]\). Otherwise \(\alpha\) is said to be transcendental.

The extension \(K / F\) is algebraic if every elements is algebraic.

Let \(\alpha\) is algebraic over \(F\), and \(M \subseteq F[x]\) be those who has \(\alpha\) as a root. then there exists a unique monic irreducible \(m_{\alpha, F}(x) \in M\), and it's the common divisor of \(M\).

(Corollary) If \(L / F\) and \(\alpha\) is algebraic over both \(L\) and \(F\), then \(m_{\alpha, L}(x) \divide m_{\alpha, F}(x)\) in \(L[x]\).

Definition. \(m_{\alpha, F}(x)\) is called the minimal polynomial for \(\alpha\) over \(F\). It's degree is called the degree of \(\alpha\).

\(\alpha\) be algebraic over \(F\) \(\implies\) \(F(\alpha) \cong F[x] / (m_{\alpha, F}(x))\).
\(\alpha\) be algebraic over \(F\) \(\implies\) \([F(\alpha) \colon F]\) is equal to the degree of \(\alpha\).
\(\alpha\) be algebraic over \(F\) \(\iff\) \([F(\alpha) \colon F]\) is finite.

Let \(F \subseteq K \subseteq L\) all be fields. Then \([L \colon F] = [L \colon K] [K \colon F]\).

Definition. An extension \(K / F\) is finitely generated if there are finite subset \(A \subseteq K\) such that \(K = F(A)\).

Definition. Let \(K_1, K_2\) be subfields of \(K\). Then the composite field of them, denoted \(K_1 K_2\), is the smallest subfield containing \(K_1 \cup K_2\).

Let \(K_1, K_2\) be finite extensions of \(F\) in \(K\), then \[ [K_1 K_2 \colon F] \le [K_1 \colon F] [K_2 \colon F] \]

In particular, when \(([K_1 \colon F], [K_2 \colon F]) = 1\), then it's equal.

splitting fields and algebraic closures

Definition.

The splitting field for \(f(x) \in F[x]\) is the smallest extension \(K / F\) such that \(f(x)\) splits completely in \(K[x]\).

\(K\) is a splitting field over \(F\) if it's the splitting field for some polynomial in \(F[x]\).

The splitting field always exists, and it's unique (every two are isomorphic).
If the degree of \(f(x)\) is \(n\), then \([K \colon F] \le n!\).
Let \(\varphi \colon F \to F'\) be an isomorphism, \(f(x) \in F[x]\), and \(f'(x) = \varphi(f(x)) \in F'[x]\). Let \(E, E'\) be splitting fields for \(f(x), f'(x)\), respectively. Then \(\varphi\) can be extended to an isomorphism \(\sigma \colon E \to E'\).
\(K\) is a splitting field of \(F\) \(\iff\) Every irreducible \(f(x) \in F(x)\) that has root in \(K\) splits over \(K\).

Definition. The field \(\Up{F}\) is called an algebraic closure of \(F\) if \(\Up{F}\) is algebraic over \(F\), and every \(f(x) \in F[x]\) splits completely over \(\Up{F}\).

\(K\) is said to be algebraically closed if \(K = \Up{K}\).

\(\Up{F}\) exists and is unique up to isomorphism.
\(\Up{\Up{F}} = \Up{F}\).
\(K = \Up{K}\) \(\iff\) every \(f(x) \in K[x]\) has root in \(K\).
(Fundamental Theorem of Algebra) \(\bbC\) is algebraically closed.
Let \(K\) be an algebraically closed and \(F\) be a subfield. Then \(\Up{F}\) is the set of elements in \(K\) that are algebraic over \(F\).

Definition. \(f(x) \in F[x]\) is called separable if it has no multiple roots over the splitting field.

Definition. A field \(K\) is called perfect if

\(\ch(K) = 0\), or

\(\ch(K) = p\), every element of \(K\) is a \(p\)-power in \(K\).

If \(\ch(F) = p \ne 0\) then \((a + b)^p = a^p + b^p\) and \((ab)^p = a^p b^p\), and so the \(p\)-power map is an injective homomorphism, called Frobenius endomorphism.

When \(F\) is finite, then the map is bijective, and so \(F\) is perfect.

\(f(x)\) is separable \(\iff\) \((f(x), D_x f(x)) = 1\).

Suppose \(F\) is perfect.

\(f(x)\) is separable \(\iff\) \(f(x)\) is the product of distinct irreducible polynomials.
\(f(x)\) is irreducible \(\implies\) \(f(x)\) is separable.

Definition. The field \(K\) is separable over \(F\) if every element in \(K\) is the root of a separable polynomial over \(F\).

\([K \colon F]\) is finite, \(F\) is perfect \(\implies\) \(K\) is separable over \(F\).

cyclotomic polynomials and extensions

TODO

Definition.

Let \(\mu_n = \{1, \zeta, \zeta^2, \ldots, \zeta^{n-1}\} \cong Z_n\) denote the group of \(n\)-th roots of \(1_{\bbQ}\).

Define the \(n\)-th cyclotomic polynomial \(\Phi_n(x)\) to be the polynomial whose roots are the primitive \(n\)-th roots of \(1_{\bbQ}\):

\[ \Phi_n(x) = \prod_{(k, n) = 1} (x - \zeta_n^k) \]

The cyclotomic polynomial \(\Phi_n(x)\) is a monic polynomial in \(\bbZ[x]\) of degree \(\varphi(n)\).

TODO

Galois Theory

automorphisms and fixed fields

Let \(K\) be a field.

Definition.

An isomorphism from \(K\) to \(K\) is called an automorphism of \(K\).

The collection of automorphism of \(K\) is denoted \(\Aut(K)\).

Suppose \(\sigma \in \Aut(K)\).

We write \(\sigma \alpha\) for \(\sigma(\alpha)\).

\(\sigma\) is said to fix an element \(\alpha \in K\) if \(\sigma \alpha = \alpha\).

\(\sigma\) is said to fix a subset \(A \subseteq K\) if \(\sigma\) fix all the elements in \(A\).

Suppose \(K / F\) is an extension.

\(\Aut(K / F)\) is the subset of \(\Aut(K)\) which fix \(F\).

Let \(K / F\) be an extension and \(\alpha \in K\), \(\sigma \in \Aut(K / F)\).

If \(f(x) \in F[x]\), then \(f(\sigma \alpha) = 0 \iff f(\alpha) = 0\).
In particular, if \(\alpha\) is algebraic, then \(m_{\alpha, F} (\sigma \alpha) = 0\).

We can see the \(\sigma \in \Aut(F(A) / F)\) just map the roots of minimal polynomail to the roots and fix \(F\).

Definition. If \(H \subseteq \Aut(K)\), the fixed field of \(H\) is the subset of \(K\) of elements fixed by the automorphisms in \(H\)

\(\Aut(K)\) is a group under composition and \(\Aut(K / F) \le \Aut(K)\).
The fixed field of \(H\) is a subfield of \(K\).

Definiton.

If \([K \colon F]\) is finite and \(|\Aut(K / F)| = [K \colon F]\), then

\(K\) is said to be Galois over \(F\)

\(K / F\) is a Galois extension.

\(\Aut(K / F)\) is called the Galois group of \(K / F\), denoted \(\Gal(K / F)\).

Example.

\(\bbQ(\sqrt{2}) / \bbQ\) is Galois, and \(\Gal(\bbQ(\sqrt{2}) / \bbQ) = \{id, \sigma\} \cong Z_2\).
\(\bbQ(\sqrt{3}) / \bbQ\) is not Galois, because \(\Aut(\bbQ(\sqrt{3}) / \bbQ) = \{id\}\).
\(\bbQ(\sqrt{2}, \sqrt{3}) / \bbQ\) is Galois, and \(\Gal(\bbQ(\sqrt{2}, \sqrt{3}) / \bbQ) = \{id, \sigma, \tau, \sigma \tau\} \cong V_4\). And
- \(\{id\}\) fix \(\bbQ(\sqrt{2}, \sqrt{3})\).
- \(\{id, \sigma\}\) fix \(\bbQ(\sqrt{3})\).
- \(\{id, \tau\}\) fix \(\bbQ(\sqrt{2})\).
- \(\{id, \sigma \tau\}\) fix \(\bbQ(\sqrt{6})\).
- \(\{id, \sigma, \tau, \sigma \tau\}\) fix \(\bbQ\).

Suppose \(F \subseteq K\) is subfield of finite degree, and \(G \le \Aut(K)\) is finite subgroup.

Denote the \(\Aut(K / F)\) by \(F^*\), the fixed field of \(G\) by \(G^*\), \([K \colon F]\) by \(D(F)\), and \(|G|\) by \(D(G)\). The relation of subfield and automorphism subgroup:

\(D(F^*) \le D(F)\).
\(D(G^*) = D(G)\).
\(F \subseteq F^{**}\).
\(G = G^{**}\).
\(F_1 \subseteq F_2 \implies F_2^* \le F_1^*\).
\(G_1 \le G_2 \implies G_2^* \subseteq G_1^*\).
\(G_1 \ne G_2 \implies G_1^* \ne G_2^*\).
\(K / F\) is Galois \(\iff\) \(D(F) = D(F^*)\) \(\iff\) \(F = F^{**}\).

splitting fields and separable polynomials

Let \(E\) be the splitting field over \(F\) for \(f(x) \in F[x]\), then \[ |\Aut(E / F)| \le [E \colon F] \] with equality if \(f(x)\) is separable over \(F\).

Definiton. If \(f(x) \in F[x]\) is separable, then the Galois group of \(f(x)\) is the Galois group of the splitting field of \(f(x)\).

The extension \(K / F\) is Galois iff \(K\) is the splitting field of some separable polynomial over \(F\).
If \(K / F\) is Galois, then \(f(x) \in F[x]\) is irreducible and has root in \(K\) \(\implies\) \(f(x)\) is separable and splits completely in \(K\).

the fundamental theorem of galois theory

Definition.

A (linear) character \(X\) of a group \(G\) with elements in field \(L\) is a homomorphism from \(G\) to the multiplicative group of \(L\): \[ X \colon G \to L^{\times}. \]

An injective homomorphism from field \(K\) to field \(L\) is called an embedding of \(K\) into \(L\).

Clearly embdding can be treated as character of \(K^{\times}\) in \(L\).

(Linear Independence of Characters)

If \(X_1, X_2, \ldots, X_n\) are distinct characters of \(G\) with elements in \(L\), then thery're linearly independent over \(L\), that is, \[ a_1 X_1 + a_2 X_2 + \cdots a_n X_n = 0, \;\; a_i \in L \] implies all \(a_i = 0\).

(Corollary)

If \(\sigma_1, \sigma_2, \ldots, \sigma_n\) are distinct embeddings of \(K\) into \(L\), then they're linearly independent over \(L\).

Definition. Let \(K / F\) be a Galois extension.

Let \(\alpha \in K\). The elements \(\sigma \alpha\) for \(\sigma \in \Gal(K / F)\) are called the conjugates of \(\alpha\) over \(F\).

Let \(F \subseteq E \subseteq K\). \(\sigma(E)\) is called the conjugate field of \(E\) over \(F\).

Recall if \(\alpha\) is a root of some polynomial in \(F[x]\), then its conjugate must also be a root. Thus we can get the following theorem.

Let \(K / F\) be Galois and \(G = \Aut(K / F)\).

\[ m_{\alpha, F} = \prod_{\sigma \in G} (x - \sigma \alpha). \]

(Fundamental Theorem of Galois Theory)

Let \(K / F\) be a Galois extension. Suppose \(E \subseteq K\) is some subfield containing \(F\), and \(H \le G = \Aut(K / F)\) is some subgroup.

Then the map \(E \mapsto H = \Aut(K / E)\) is a bijection, which inverse is \(H \mapsto E\), the fixed field of \(H\). We can deduce that

\([K \colon E] = |H|\).
\([E \colon F] = [G \colon H]\).
\(E_1 \subseteq E_2 \iff H_2 \le H_1\).
\(E / F\) is Galois \(\iff\) \(H \unlhd G\).
\(E / F\) is Galois \(\implies\) \(\Gal(E / F) \cong G / H\).
\(E_1 \cap E_2\) corresponds to \(\Inner{H_1, H_2}\).
\(E_1 E_2\) corresponds to \(H_1 \cap H_2\).

finite fields

(Recall) Every finite field has order \(p^n\), and is unique up to isomorphism. We Denote it by \(\bbF_{p^n}\).
\(\bbF_{p^n}\) is the splitting field over \(\bbF_p\) for \(x^{p^n} - x\), the elements are the roots.
\(\Gal(\bbF_{p^n} / \bbF_p) \cong Z_n\), generated by \(\sigma_p \colon \alpha \mapsto \alpha^p\).

We show althought \(x^4 + 1\) is irreducible in \(\bbZ[x]\), it is reducible in every \(\bbF_p\). If \(p = 2\) we have \(x^4 + 1 = {(x + 1)}^4\) and otherwise we have \[ x^4 + 1 \divide x^8 - 1 \divide x^{p^2 - 1} - 1 \divide x^{p^2} - x \] Thus all its roots is contained in \(\bbF_{p^2}\). Hence the extension generated by any root of \(x^4 + 1\) is at most of degree \(2\), and so \(x^4 + 1\) cannot be irreducible.

(Recall) multiplicative group of finite field is always cyclic.

(Corollary) \(\bbF_{p^n}\) is always a simple extension of \(\bbF_p\). \(\bbF_{p^n} = \bbF_p(\theta)\) where \(\theta\) is an generator of the multiplicative group.

\(x^{p^n} - x\) is the product of all the distinct irreducible polynomials in \(\bbF_p[x]\) of degree \(d\) where \(d\) runs through all divisors of \(n\).

composite extensions and simple extensions

Suppose \(K / F\) is Galois and \(F' / F\) is arbitrary, then \(K F' / F'\) is Galois, and \(\Gal(K F' / F') \cong \Gal(K / K \cap F')\).

(Corollary) \[ [K F' \colon F] = \frac{[K \colon F] [F' \colon F]}{[K \cap F' \colon F]} \]

Let \(K_1, K_2\) be Glaois extensions of \(F\).

\(K_1 \cap K_2 / F\) is Glaois.
\(K_1 K_2 / F\) is Glaois.

If \(K_1 \cap K_2 = F\) we have \[ \Gal(K_1 K_2 / F) \cong \Gal(K_1 / F) \times \Gal(K_2 / F). \]

Recall \(K / F\) is simple if \(K = F(\theta)\), in this case \(\theta\) is called a primitive element for \(K\).

(The Primitive Element Theorem) Every finite and separable extension is simple.

(Recall) \([K \colon F]\) is finite, \(F\) is perfect \(\implies\) \(K\) is separable over \(F\).

Thus every finite extension of perfect field is simple.

Galois groups of polynomials

Let \(x_1, x_2, \ldots, x_n\) be indeterminates.

Definition.

The elementary symmetric functions \(s_1, s_2, \ldots, s_n\) are defined by \[ \begin{aligned} s_1 &= x_1 + x_2 + \cdots + x_n \\ s_2 &= x_1 x_2 + x_1 x_3 + \cdots + x_2 x_3 + x_2 x_4 + \cdots + x_{n-1} x_n \\ &\vdots \\ s_n &= x_1 x_2 \cdots x_n \end{aligned} \]

The general polynomial of degree \(n\) is \[ (x - x_1) (x - x_2) \cdots (x - x_n) \]

A rotional function of \(\List{n}{x}\) is symmetric iff is not changed by any permutation of \(\List{n}{x}\).

Notice any rotional function of \(\List{n}{x}\) can be treated as an elements in \(F(\List{n}{x})\).

\(F(\List{n}{x}) / F(\List{n}{s})\) is Galois.
In \(F(\List{n}{x})\), the fixed field of \(S_n\) is exactly \(F(\List{n}{s})\).
Every symmetric rotional function of \(\List{n}{x}\), is some rotional function of \(\List{n}{s}\).
The general polynomial over \(F(\List{n}{s})\) is separable with Galois group \(S_n\).

Definition.

The discriminant of \(\List{n}{x}\) is \[ D = \prod_{i < j} (x_i - x_j)^2. \]

The discriminant of polynomial is the discriminant of the roots.

We define \(\sqrt{D} = \prod_{i<j} (x_i - x_j)\).

If \(\ch(F) \ne 2\) then: \(\sigma \in A_n\) \(\iff\) \(\sigma\) fixed \(\sqrt{D}\).

Suppose \(D\) is the discriminant of some polynomial \(f(x) \in F[x]\).

\(f(x)\) is separable \(\iff\) \(D \ne 0\).
\(D \in F\).
If \(D \ne 0\): the Galois group of \(f(x)\) is a subgroup of \(A_n\) iff \(\sqrt{D} \in F\).

the funamental theorem of algebra

Every polynomial \(f(x) \in \bbC[x]\) of degree \(n\) has precisely \(n\) roots in \(\bbC\).

Two simple facts:

Every polynomial in \(\bbR[x]\) of odd degree has a root in \(\bbR\).
Quadratic polynomials \(f(x) \in \bbC[x]\) have all roots in \(\bbC\).

Suppose \(f(x)\) is of degree \(n = 2^k m\) where \(m\) is odd. We prove \(f(x)\) has a root in \(\bbC\) by induction on \(k\).

For \(k = 0\) it's the simple fact. Suppose now \(k \ge 1\). Let \(\List{n}{\alpha}\) be roots of \(f(x)\) and \(K = \bbR(\List{n}{\alpha}\). Then \(K\) is a Galois extension of \(\bbR\) containing \(\bbC\) and \(\List{n}{\alpha}\).

For any \(t \in \bbR\) we define a polynomial \[ L_t(x) = \prod_{1 \le i < j \le n} (x - l\alpha_i - \alpha_j - t \alpha_i \alpha_j) \] Then any \(\sigma \in \Aut(K / \bbR)\) permmute the terms in this product, and so \(L_t(x)\) is fixed by \(\Gal(K / \bbR)\). Hence \(L_t(x) \in \bbR[x]\) of degree \(n (n-1) / 2 = 2^{k-1} m'\) for some odd integer \(m'\).

By induction hypothesis \(L_t(x)\) has a root in \(\bbC\). Hence for any \(t \in \bbR\) we can find some \(i, j\) such that \(\alpha_i + \alpha_j + t \alpha_i \alpha_j \in \bbC\).

Thus there exists some \(i, j\) such that we can find two distinct real number \(s, t \in \bbR\) satisfying \(\alpha_i + \alpha_j + s \alpha_i \alpha_j \in \bbC\) and \(\alpha_i + \alpha_j + t \alpha_i \alpha_j \in \bbC\). It follows that \(\alpha_i + \alpha_j, \alpha_i \alpha_j \in \bbC\).

Every polynomial \(f(x) \in \bbR[x]\) has a root in \(\bbC\).

solvable and radical extensoins

Definition. The extension \(K / F\) is said to be cyclic if it's Galois with a cyclic Galois group.

Suppose \(\ch(F) \ndivide n\) and \(F\) contains the \(n\)-th roots of unity \(\zeta\).

\(F(\sqrt[n]{a}) / F\) is cyclic for all \(a \in F\).
Any cyclic extension is \(F(\sqrt[n]{a})\) for some \(a \in F\).
\([F(\sqrt[n]{a}) \colon F] \divide n\).
Define the Lagrange resolvent \((\alpha, \zeta) \in K\) by \[ (\alpha, \zeta) = \sum_{k=0}^{n-1} \zeta^k \sigma^k (\alpha), \] where \(\sigma \in \Gal(K / F)\) and \(\alpha \in K\).