Abstract Algebra I
 Level: 0 1 2 3 4 5 6 7
MSC Classification: 20, 12 (Group theory, Field theory and polynomials)

# Getting Oriented

 Rough Guides to Algebra
 General Algebra Special Topics

The main goal in abstract algebra is extending the operations and properties we take for granted on sets we're used to working with (like integers, reals, complex numbers, etc.) to arbitrary sets. This requires precise definitions and requirements on the structure of the set in order to ensure the desired properties are present.

The starting point of abstract algebra is the group, which is just a set with an operation such as addition. A surprising number of sets fit into this category, allowing us to analyze things as diverse as the integers modulo k, invertible matrices, and symmetries of a polygon all at once. We'll eventually turn to the question of classification of groups, which is done both by size and by type.

Later, we will encounter the ring, which is just a set with two operations (traditionally addition and multiplication) such as the real numbers. A related structure is the field, which also has two operations but allows for division as well. Several more definitions fall somewhere between rings and fields with specific defining properties, such as the existence of a division algorithm. Our last goal will be proving the insolvability of the quintic, a subject which demonstrates a strong relationship between groups and fields.

Groups, rings, and fields are the first structures usually encountered in abstract algebra, and the definitions intended to make them look like known sets give rise to a much more general theory. More definitions are made to accommodate this theory than in perhaps any other field of mathematics. Memorizing the most basic definitions is recommended, and will prove to make the rest more palatable.

# Groups

## The Basics

Without further ado, here is the definition of a group:

Definition

A group is a set G with a binary operation * that satisfies the following axioms:

1. Associativity axiom: $(a*b)*c=a*(b*c)$ for all $a,b,c\in G$
2. Identity axiom: there is an element $e\in G$ such that $a*e=e*a$ for all $a \in G$
3. Inverse axiom: for each $a\in G$, there is an element $a^{-1}\in G$ such that $a*a^{-1}=a^{-1}* a=e$.

If, in addition, the operation is commutative ($a*b=b*a$ for all $a,b \in G$), then the group is abelian.

The group is denoted $(G,*)$ if one wishes to specify the operation explicitly. Typical examples of group operation include addition and multiplication. Subtraction and division are usually not group operations since they are not associative.

Some essential group properties include uniqueness of the identity, left and right cancellation laws, and uniqueness of inverses.

The order of a group is the number of elements it contains. Groups may be finite or infinite. The order of an element $a\in G$ is the smallest $k\in\mathbb{Z}^+$ such that $a^k=e$, where exponentiation denotes repeated operation. An element has infinite order if no such integer exists.

### Examples

• Sets that are groups under addition include the integers $\mathbb{Z}$, the rationals $\mathbb{Q}$, the reals $\mathbb{R}$, complex numbers $\mathbb{C}$, vectors, and matrices.
• The set $\mathbb{Z}_n=\{0,1,2,\ldots,n-1\}$ becomes a group under addition modulo n.
• Groups under multiplication include the rationals, the reals, and complex numbers, when 0 is omitted.
• Invertible matrices form a group under matrix multiplication.
• The complex roots of unity $\{e^{2\pi i k/n}\}_{i=0}^{n-1}$ form a group under the operation of complex multiplication. This is actually the same group as addition modulo n.
• The symmetries of a polygon with n sides form the dihedral group $D_n$. It has $2n$ elements: n rotations and n reflections.
• The positive integers less than n that are relatively prime to n form a group under multiplication, denoted $U(n)$. The order of this group defines the Euler phi function: $\phi(n)=|U(n)|$.

## Subgroups and Cyclic Groups

Definition

A subgroup H of a group G is a subset that is itself a group under the operation of G. One writes H < G to indicate this relationship.

For example, under addition one has a chain of subgroups $\mathbb{Z} < \mathbb{Q} < \mathbb{R} < \mathbb{C}$.

A subset is a subgroup if and only if it is closed under both the group operation and inversion. For a finite group, one only needs to test for closure under the group operation.

Here are a few ways to construct a subgroup (in the notation of multiplicative groups):

• The cyclic subgroup $\langle{a}\rangle$ generated by a is $\{e,a,a^{-1},a^2,a^{-2},\ldots,\}$.
• The center $Z(G)$ of G consists of the elements of G that commute with all other elements.
• The centralizer $C(a)$ of $a\in G$ consists of the elements of G that commute with a.

The subgroups $\langle{a}\rangle$ and $Z(G)$ are always abelian. In general, a cyclic group is one generated by a single element (and its inverse). These groups are all abelian and behave like either the integers $\mathbb{Z}$ or the finite group $\mathbb{Z}_n$. Some elementary properties of cyclic groups follow, with $G=\langle{a}\rangle$ having order n:

• if $a^k=e$, then k|n (k divides n);
• $G=\langle{a^k}\rangle$ if and only if $\mathrm{gcd}(n,k)=1$ (these are the generators of the group);
• each divisor $k$ of n corresponds to one subgroup of order k, namely $\langle{a^{n/k}}\rangle$;
• the number of elements of order n is given by $\phi(n)$, the Euler phi function.

## Cosets and Factor Groups

Any subgroup naturally partitions a group into disjoint subsets, called cosets:

Definition

A left coset of a subgroup $H<G$ is a subset $aH=\{ah:h\in H\}$ for some $a\in G$, while a right coset is given analogously by $Ha=\{ha:h\in H\}$.

The fact that the cosets partition the group, and also that $aH=H$ if and only if $a\in H$, follows from:

Lagrange's Theorem

The index $[G:H]=|G|/|H|$ of any subgroup is an integer, equal to the number of distinct left (or right) cosets of H.

It follows immediately that all groups of prime order are cyclic (since they have no nontrivial subgroups), that $a^{|G|}=e$ for all elements $a\in G$, and that $a^p \equiv a \mod p$ for p prime (Fermat's Little Theorem).

When the left and right cosets of a group always coincide, the set of cosets has a group structure:

Definition

A normal subgroup H is one with $aH=Ha$ for all $a\in G$, or equivalently $aH a^{-1}=H$. Normalcy is denoted by $H\lhd G$. The set of cosets of a normal subgroup form the factor group (or quotient group) $G/H$, with operation inherited from the group.

By Lagrange's Theorem, $|G/H|=|G|/|H|$. Elements of a factor group $G/H$ are usually denoted gH for some $g\in G$.

## Group Morphisms

Any "interesting" map between groups should preserve the group operation:

Definition

A homomorphism from one group G to another group G' is a map $\phi:G\to G'$ that preserves the group operation, so that $\phi(ab)=\phi(a)\phi(b)$.

Homomorphisms preserve many properties of a group, taking (abelian or normal) subgroups to (abelian or normal) subgroups. The same is true for the inverse image of a homomorphism.

The notion of group equivalence arises when the map is also bijective.

Definition

An isomorphism is a bijective homomorphism $\phi:G\to G'$ whose inverse $\phi^{-1}:G'\to G$ is also a homomorphism. The groups G and G' are isomorphic, and we write $G\cong G'$.

Every property of a group (other than the names of its elements), is preserved under isomorphism. A fundamental question is classifying non-isomorphic groups. Proving that two groups are not isomorphic is usually not too hard: one just needs to find properties of the two that differ. But proving that two groups are isomorphic requires one to identify a specific isomorphism.

An automorphism is an isomorphism from a group to itself, and such maps form the automorphism group $(\mathrm{Aut}(G),\circ)$ under function composition. Maps of the form $\phi_a(x)=axa^{-1}$ for some $a\in G$ are always automorphisms and form the inner automorphism group $\mathrm{Inn}(G)<\mathrm{Aut}(G)$. As a quick example, $\mathrm{Aut}(\mathbb{Z}_n)\cong U(n)$. Homomorphisms from G to another group G' also form a group, denoted $\mathrm{Hom}(G,G')$, with group operation inherited from the operation on G'.

The center of a group $Z(G)$ is always normal, and the factor group $G/Z(G)$ is isomorphic to the inner automorphism group $\mathrm{Inn}(G)$. In particular, it is cyclic if and only if G is abelian (the G / Z Theorem), and if G is abelian then $Z(G)=G$ so $G/Z(G)$ is trivial. This can be used to show that an abelian group G has elements of each prime order dividing $|G|$.

## Permutation Groups

A permutation group is a group of bijective maps from a set A to itself under the group operation of composition. A first example is the automorphisms $\mathrm{Aut}(G)$. Permutation groups are key in understanding groups:

Cayley's Theorem

Every group is isomorphic to a subgroup of a permutation group.

To specify a permutation, one must clearly show where each element of the set A is taken. In the finite case, one can use the following notation:

(1)
\begin{align} \alpha=\begin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6\\ 2 & 3 & 1 & 6 & 5 & 4\end{bmatrix}. \end{align}

This indicates that 1 is mapped to 2, 2 to 3, and so on.

A more compact notation is cycle notation. This would give the above as $(1\:2\:3)(4\:6)$, which means swap 4 and 6, and then rotate 1, 2, and 3. As for function compositions, the process starts on the right and works left. The permutation can also be written as $(2\:3)(1\:3)(4\:6)$. Disjoint cycles commute, and all permutations can be written as a product of disjoint cycles. The order of the permutation is the least common multiple of the cycle lengths.

The group of all $n!$ permutations on n elements is known as the symmetric group $S_n$. Every finite permutation can be written as the product of transpositions, or 2-cycles, and the number of such cycles always has the same parity. Thus, we can speak of even or odd permutations. The even permutations form the Alternating Group An, which has exactly $n!/2$ elements.

Given a group of permutations G on a set A, each element $a\in A$ determines a subgroup of permutations called the stabilizer subgroup, denoted by $\mathrm{stab}(a)$ or Ga, which consists of the permutations that fix a. The element also determines a subset of A called the orbit, denoted by $\mathrm{orb}(a)$ or Ga, which consists of all the elements mapping to a under permutations in the group. The Orbit-Stabilizer Theorem says that $|G|=|\mathrm{orb}(a)||\mathrm{stab}(a)|$ for any $a\in A$.

## The Isomorphism Theorems

In this section we look at three theorems that are also true, with slight modifications, for rings and other algebraic structures besides groups.

The kernel $\ker\phi$ of a homomorphism $\phi:G\to G'$ is the set of elements of G that map to the identity: $\ker\phi = \{g\in G : \phi(g)=e\}$. The image of the homomorphism will be denoted $\phi(G)$.

First Isomorphism Theorem

For any group homomorphism $\phi:G\to G'$, there is an isomorphism $G/\ker\phi \cong \phi(G)$ given by $g\:\ker\phi\mapsto\phi(g)$.

This is somewhat intuitive: it says that the image of a homomorphism looks like a factor group of G with the elements mapping to the identity being trivial.

A corollary is the N/C Theorem, which states that $N(H)/C(H)$ is isomorphic to a subgroup of $\mathrm{Aut}(H)$. (Recall that $N(H)$ is the normalizer of H, the set of elements $g\in G$ with $gH g^{-1}\in H$, and $C(H)$ is the centralizer of H, the elements of G commuting with all elements of H). The isomorphism takes an element $g\in G$ to the inner automorphism $\phi_g:h\mapsto gh g^{-1}$, which is a group homomorphism with kernel $C(H)$.

A simple application of this result is the proof that every group of order 35 is cyclic. First, counting arguments show that G must have elements of order both 5 and 7. Let H be a cyclic subgroup of order 7. One can show that $N(H)=G$ and that $H<C(H)$. But $|N(H)/C(H)|$ must divide 6, the order of $\mathrm{Aut}(H)$, which is possible only if $|C(H)|=35$. Then, an element hk, where $h\in H$ is nontrivial and k has order 5, will generate the group.

The second and third isomorphism theorems have a similar flavor:

• Second Isomorphism Theorem: Given a subgroup $K\leq G$ and a normal subgroup $N\lhd G$, $K/K\cap N \cong KN/N$.
• Third Isomorphism Theorem : Given subgroups $N\lhd M\lhd G$, $\frac{G/N}{M/N}\cong G/M$.

## Direct Products and Finite Abelian Groups

Having analyzed subgroups and factor groups, which in a sense make a group smaller, we now look at direct products, which form bigger groups out of smaller ones. The simplest product group is the external direct product of groups G and G', denoted $G\oplus G'$ and defined as the set of elements $(g,g')$ with operation acting separately on each coordinate. $(G,e')\cong G$ and $(e,G')\cong G'$ are then subgroups.

The order of an element $(g,g')$ is the least common multiple of the orders of g and g'. Thus, a group $G\oplus G'$ is cyclic if and only if both G and G' are cyclic, with $|G|$ and $|G'|$ relatively prime. In particular, for $(m,n)=1$, we have $\mathbb{Z}_m\oplus \mathbb{Z}_n=\mathbb{Z}_{mn}$ and $U(m)\oplus U(n)=U(mn)$.

The internal direct product is a way of representing a given group as a direct product of its subgroups; we write $G=H\times K$ if $G=HK$ where H and K are normal subgroups and $H\cap K=\{e\}$. Since $H\oplus K \cong H\times K$, the only difference between the products in this case is that the external sum builds bigger groups from smaller ones, while the internal sum breaks groups into smaller pieces.

Finite abelian groups are completely classified by the following theorem:

Fundamental Theorem of Finite Abelian Groups

Every finite abelian group is isomorphic to a unique direct product of cyclic groups whose orders are prime powers, that is $G\cong \mathbb{Z}_{p_1}^{n_1}\oplus \cdots\oplus \mathbb{Z}_{p_k}^{n_k}$.

This fundamental result usually caps one's first exposure to groups. As a corollary, such a group has subgroup of order m for each m dividing the group order.

# Rings

## The Basics

Having studied sets with one operation, it is time to move on to sets with two operations, called rings.

Definition

A ring is a set R with two operations + and * (usually suppressed) such that:

1. R is an abelian group under +;
2. R is associative under *;
3. R is distributive, meaning $a*(b+c)=a*b+a*c$.

The additive identity is denoted 0. If $ab=ba$, R is a commutative ring, and if there is a multiplicative identity 1, R is a ring with unity. A commutative ring with unity which is an abelian group under *, excluding 0, is called a field.

In a ring with unity, elements with multiplicative inverses are called units, so in a field every element but 0 is a unit. Inverses and identities, if they exist, are unique.

The characteristic of a ring R is the smallest integer n such that $na=0$ for all $a\in R$, or 0 if no such exists. If the ring has a unity, the characteristic is the order of the unity (or 0 if it has infinite order).

Subrings are subsets of rings that are themselves rings; equivalently they are closed under subtraction and multiplication. For example, $\mathbb{Q}$ is a subring of $\mathbb{R}$, and $2\mathbb{Z}$ is a subring of $\mathbb{Z}$.

### Examples

• The integers $\mathbb{Z}$ are a commutative ring with unity. The units are +1 and -1.
• The even integers $2\mathbb{Z}$ are a commutative ring without unity.
• The integers modulo n, $\mathbb{Z}_n$, are a commutative ring with unity. The units are $U(n)$.
• Integer $2\times2$ matrices are a noncommutative ring with unity $\binom{1\:0}{0\:1}$.

The example $\mathbb{Z}_n$ above has characteristic n, while the remaining examples have characteristic 0.

### Polynomial Rings

Given a ring R, the ring of polynomials in $\diamondsuit$ with coefficients in R is $R[\diamondsuit] = \{a_0+a_1\diamondsuit+\cdots+a_k\diamondsuit^k : a_0, a_1, \ldots, a_k \in R\}$. Elements of $R[\diamondsuit]$ are sometimes denoted as "functions" $f(\diamondsuit)$, but it is important to note that they are not actual functions; they could just as easily be described by the coefficients only: $[a_0,a_1,\ldots,a_k]$.

The ring of polynomials is itself a ring, and R is one of its subrings. In some cases, one assumes that $\diamondsuit^k \in R$ for some k, so the only polynomials of interest have small order. For instance, the Gaussian Integers are $\mathbb{Z}[i]$, where it is assumed that $i^2=-1$.

## Ideals and Factor Rings

With groups, we needed a normal subgroup to form a factor group; with rings, we need a:

Definition

A (2-sided) ideal is a subring A of R such that $ra,ar\in A$ for all $a\in A, r\in R$. This gives a well-defined factor ring $R/A$, consisting of the cosets of the ideal A, which are typically denoted $r+A$ for some $r\in R$.

A simple example is $\mathbb{Z}/4\mathbb{Z}$, consisting of the elements $\{0,1,2,3\}$ with addition and multiplication modulo 4.

Any single element $a\in R$ generates a principal ideal, defined as $\langle{a}\rangle=\{ar,ra:r\in R\}$. For example, the factor ring $\mathbb{R}[x]/\langle{x^2+1}\rangle$ is defined using a principal ideal. It can be thought of as polynomials with real coefficients and the relation $x^2=-1$, and is thus equivalent to the complex ring $\mathbb{C}$.

A prime ideal $A\subset R$ is one in which $ab\in A$ implies either $a\in A$ or $b\in B$. This is a similar notion to prime numbers; indeed, $\mathbb{Z}/n\mathbb{Z}$ is a prime ideal of $\mathbb{Z}$ if and only if n is prime. A maximal ideal is a proper ideal A such that if $A\subseteq B \subseteq R$ for another ideal B, then either $B=A$ or $B=R$. The ideal $\langle{x^2+1}\rangle$ in $\mathbb{R}[x]$ is an example of a maximal ideal. These ideals are important because $R/A$ is a field if and only if A is maximal, and $R/A$ is an integral domain (the topic of the next section) if and only if A is prime.

## Integral Domains

An integral domain is one of many types of rings with a specific structure, in this case meant to ensure behavior like the integers (hence the name).

Definition

An integral domain is a commutative ring with unity and no zero divisors (elements with $ab=0$). Equivalently, the cancellation laws $ab=ac \Rightarrow b=c$ and $ba=ca \Rightarrow b=c$ hold.

Integral domains have characteristic 0 or p prime.

Examples of integral domains include $\mathbb{Z}/3\mathbb{Z}[i]$ and $\mathbb{Q}[\sqrt{2}]$ (recall this notation from the previous section). The first is also a finite field, since all finite integral domains are fields. In general, the structure of finite fields is very well understood.

For every integral domain D, there is a field F with D as a subring, called the field of quotients. This is the analog of the rationals for integers. As can be expected, F is constructed by looking at all pairs (quotients) of elements of D and an appropriate equivalence relation.

The study of polynomials is central to ring theory. A ring of polynomials $D[x]$ over an integral domain D is itself an integral domain. Working in an integral domains allows us to say that if $(x-a)(x-b)=0$, then either $x=a$ or $x=b$; this is not always the case in a ring with zero divisors. If D is a field, then one has a division algorithm in $D[x]$, and polynomials are factored in the usual manner.

A principal ideal domain, or PID, is an integral domain in which every ideal is a principal ideal (generated by a single element). The polynomials $F[x]$ over a field form a PID. In fact, an ideal I of $F[x]$ is generated by the polynomials of minimal degree in I.

Other than the special elements 0 and 1, integral domains also lack nilpotent elements (those satisfying $a^n=0$ for some n) and idempotent elements (those satisfying $a^2=a$).

## Ring Morphisms

With rings, homomorphisms and isomorphisms are defined in a way similar to groups. The main difference is that ring homomorphisms preserve both group operations.

An example for polynomial rings is the evaluation homomorphism $\mathbb{R}[x]\to\mathbb{R}$ defined by $f(x)\mapsto f(1)$.

Analogous to the situation for groups, kernels of homomorphisms are ideals, and all ideals are kernels of some ring homomorphism. We also have:

First Isomorphism Theorem for Rings

Given a ring homomorphism $\phi:R\to S$, there is an isomorphism $R/\mathrm{ker}\:\phi\cong\phi(R)$ given by the map $r+\mathrm{ker}\phi\mapsto\phi(r)$.

Thus, for example, $\mathbb{Z}[x]/\langle{x}\rangle\cong\mathbb{Z}$. Since $\mathbb{Z}$ is an integral domain but not a field, this means that $\langle{x}\rangle$ is a prime ideal, but not maximal.

Another important homomorphism is the map $\phi:\mathbb{Z}\to R$ taking $n\mapsto 1+\cdots+1$, where the unity $1\in R$ is added to itself n times. Since this is a ring homomorphism, every ring with unity has a subring isomorphic to either $\mathbb{Z}$ or $\mathbb{Z}/n\mathbb{Z}$, and the choice between these is determined by the characteristic of the ring. Analogously, a field always has a subfield isomorphic to either $\mathbb{Q}$ or $\mathbb{Z}/n\mathbb{Z}$, again depending on the characteristic. This subfield, called the prime subfield, is actually the intersection of all nontrivial subfields.

## More on Polynomial Rings

Polynomials are widely studied in ring theory because they are the simplest generalization of the integers, with many properties carrying over.

Definition

Given an integral domain D, an irreducible polynomial $f(x)\in D[x]$ is one for which $f(x)=g(x)h(x)$ implies either g or h is a unit. Otherwise, the polynomial is reducible.

A primitive polynomial $f(x)\in\mathbb{Z}[x]$ is one with no common factor among its coefficients. The product of two primitive polynomials is also primitive (the Gauss Lemma). This property is frequently used to simplify proofs.

In a field F, a polynomial of degree 2 or 3 is reducible if and only if it has a zero in F. If a polynomial in $\mathbb{Z}[x]$ is reducible over $\mathbb{Q}$, then it is also reducible over $\mathbb{Z}$. Equivalently, if it is irreducible over $\mathbb{Z}$, then it is irreducible over $\mathbb{Q}$. Hinting at some properties we'll see later, every polynomial of degree greater than 1 is reducible over some (extension) field. Actually, every polynomial $\mathbb{Z}[x]$ can be uniquely factored into a product of nonunits of degree 0 and irreducible polynomials, similar to the unique factorization of integers.

Here are a few tests for irreducibility over $\mathbb{Z}[x]$. First, given a prime p, a polynomial $f(x)\in\mathbb{Z}[x]$ gives a unique polynomial $\bar f(x)\in\mathbb{Z}/p\mathbb{Z}[x]$, found by reducing the coefficients modulo p. If this polynomial has the same degree and is irreducible over $\mathbb{Z}/p\mathbb{Z}$, then $f(x)$ is irreducible over $\mathbb{Z}$. Second, the Eisenstein criterion states that if $f(x)=a_nx^n+\cdots+a_0\in\mathbb{Z}[x]$ and some prime p divides $a_0,\ldots,a_{n-1}$ but $p\not{|}a_n$ and $p^2\not{|}a_0$, then f is irreducible over $\mathbb{Q}$. For example, when p is prime the cyclotomic polynomial $\Phi_p(x)=x^{p-1}+\cdots+x+1$ is irreducible over $\mathbb{Q}$.

In a field, $p(x)$ is irreducible if and only if $\langle{p(x)}\rangle$ is a maximal ideal. In this case, $F[x]/\langle{p(x)}\rangle$ is also a field. This provides a general technique for cosntructing fields. A simple consequence of this result is that for irreducible $p(x)|a(x)b(x)$, one must have either $p(x)|a(x)$ or $p(x)|b(x)$.

## General Integral Domains

Many properties of integers and polynomials carry over to general integral domains. In place of prime numbers/irreducible polynomials, an irreducible element $a\in D$ is one for which $a=bc$ implies either b or c is a unit; a prime element $a\in D$ is one for which $a|bc$ implies either $a|b$ or $a|c$ (which happens if and only if $\langle{a}\rangle$ is a prime ideal). All prime elements are irreducible, but the converse is not true. In $\mathbb{Z}[\sqrt{-3}]$, the element $1+\sqrt{-3}$ is irreducible but not prime. However, in a PID, elements are irreducible if and only if they are prime.

We now introduce a few more classes of integral domains. Unique factorization holds in all PIDs; a general integral domain with this property is called a Unique Factorization Domain, or UFD. Thus, for example, unique factorization holds in $F[x]$ for F a field (and if D is a UFD, then $D[x]$ is also a UFD). The proof that every PID is a UFD uses the fact that a strictly increasing chain of ideals $I_1\subset I_2\subset\cdots$ in a PID must be finite (this property in general is what characterizes a Noetherian Domain).

Another class of PID is the Euclidean domain, basically an integral domain with a division algorithm. Precisely, in a Euclidean domain D, there is a function $d:D^*\to\mathbb{Z}_0^+$ from the nonzero elements to the nonnegative integers such that whenever $d(a)\leq d(ab)$ and $b\neq 0$ there are elements $q,r\in D$ such that $a=bq+r$ and $d(r)<d(b)$. The obvious prototype is $\mathbb{Z}$, with $d(a)=|a|$. A less obvious example is $F[x]$ with $d(f(x))=\deg{f(x)}$.

One may summarize the inclusions described above by the diagram

(2)
\begin{align} \text{field} \:\Rightarrow\: \text{Euclidean domain} \:\Rightarrow\: \text{PID} \:\Rightarrow\: \text{UFD} \:\Rightarrow\: \text{integral domain} \:\Rightarrow\: \text{commutative ring with unity}. \end{align}

# Fields

Definition

A field is a set F with two operations + and * such that both $(F,+)$ and $(F,*)$ (sometimes denoted $F^*$) are abelian groups and the distributive property is satisfied.

Important areas of study in field theory include vector spaces, which are most generally defined over arbitrary fields, the study of finite fields, which are relatively easy to classify, and Galois theory, which was originally used to study the roots of polynomial equations.

## Vector Spaces

Vector spaces may be defined over arbitrary fields.

Definition

A vector space V (over a field F) is a space with a map $F\times V\to V$ such that for $a,b\in F$ and $u,v\in V$ one has $a(u+v)=au+av$, $(a+b)v=av+bv$, $a(bv)=(ab)v$, and $1v=v$.

Elements of F are the scalars, and elements of V are the vectors. A few examples include $\mathbb{R}^n$, the matrix group $M_2(\mathbb{Q})$, and $\mathbb{Z}/p\mathbb{Z}[x]$ for p prime (its basis is $\{1,x,\ldots,x^{p-1}\}$). Notions such as subspace, linear dependence/independence, basis, dimension, linear combination, and span carry over from linear algebra. It is true in general that all bases for a vector space have the same number of elements, although that number might be infinite.

## Extension Fields

Definition

An extension field E of F a field including E as a subfield. The relationship is denoted E > F. Given a polynomial $f(x)\in F[x]$, a splitting field for f is an extension field E > F that contains all the zeros of f.

The condition defining a splitting field ensures that f may be factored into a product of linear factors, hence it "splits". An important example is the extension $\mathbb{Q}[i] > \mathbb{Q}$, which is a splitting field for $f(x)=x^2+1$. In the more general context of $\mathbb{R}[x]$, the same polynomial splits over $\mathbb{C} > \mathbb{R}$.

Fundamental Theorem of Field Theory

Given a field F and a polynomial $f(x)\in F[x]$, there exists an extension field E > F in which f has a zero.

Using this theorem, it makes sense to define $F(a_1,\ldots,a_n)$ to be the smallest extension of F containing all ai. Note that the parentheses here distinguish the result as a field rather than a ring of polynomials. This extension is precisely the splitting field for the polynomial $f(x)=\prod (x-a_i)$. Some properties of splitting fields follow:

• Every polynomial has a splitting field.
• Splitting fields are unique up to isomorphism.
• Given an irreducible polynomial $p(x)\in F[x]$, with a zero a in an extension field of F, then $F(a)\cong F[x]/\langle{p(x)}\rangle$, and the elements of $F(a)$ can be expressed uniquely as polynomials of degree less than $\deg{p(x)}$.
• In the previous case, if a and b are both zeros of f, then $F(a)\cong F(b)$.

## Algebraic Extensions

Definition

Given a field extension E > F, an element $a\in E$ is algebraic over F if it is a zero of a polynomial in $F[x]$. Otherwise it is transcendental.

Standard examples are $\sqrt{2}$, an algebraic element over $\mathbb{Q}$, and$\pi$, which is transcendental. If every element of an extension field E is algebraic, E is an algebraic extension.

An extension $F(a)$ generated by a single element is a simple extension. Actually, all finite extensions over a
field of characteristic 0 are simple. An element a of an extension E > F that generates E, meaning $E\cong F(a)$, is called a primitive element.

If a is transcendental, then $F(a)\cong F(x)$, the field of quotients. Otherwise, a is algebraic and $F(a)\cong F[x]/\langle{p(x)}\rangle$, where $p(x)$ is irreducible over F and $p(a)=0$. In fact, the polynomial p is unique if taken to be monic, and it divides any other polynomial $f(x)$ with $f(a)=0$.

Every extension E > F has a subfield consisting of all algebraic elements over F, called the algebraic closure of F in E. An algebraically closed field is one with no proper algebraic extensions, such as $\mathbb{C}$, so that every polynomial has its zeros in the field.

The degree of an extension E > F is the dimension of E considered as a vector space over F, or $\infty$ if the extension is transcendental. It is denoted $[E:F]$. For example, the degree of $F(a)\supset F$ is the degree of the minimal polynomial. For $K\supset E\supset F$, the degrees satisfy $[K:F]=[K:E][E:F]$.

## Finite Fields

Finite fields are as easy to classify as finite abelian groups. For each prime p and integer n, there is a unique field $GF(p^n)$ of order pn, called the Galois field, and these are the only finite fields. As an additive group, $GF(p^n)\cong \mathbb{Z}/p\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/p\mathbb{Z}$, and as a multiplicative group, $GF(p^n)^*$ is cyclic and isomorphic to $\mathbb{Z}/(p^n-1)\mathbb{Z}$. The degree of $GF(p^n)$ over $GF(p)$ is n, and a generator of $GF(p^n)^*$ is algebraic over $GF(p)$ with degree n. The field $GF(p^n)$ has a unique subfield of order pm for every m|n, and these are the only subfields. An example is $GF(16)$, which may be constructed as $\{a_3x^3+a_2x^2+a_1x+a_0+\langle{x^4+x+1}\rangle: a_i\in\mathbb{Z}_2\}$, with generator x.

## Galois Theory

Galois theory concerns the relationship between extensions of fields and groups. It was originally used to prove the insolvability of the quintic, i.e., the nonexistence of an algebraic formula giving the roots of a fifth-order polynomial.

Definition

Given a field extension E > F, the Galois group $\mathsf{Gal}(E/F)$ is the group of automorphisms of E that fix every element of F.

We also define the fixed field $E_H$ of a group $H<\mathsf{Gal}(E/F)$ as the set of elements $x\in E$ fixed by every automorphism in H.

There is a remarkable relationship between the subgroups of $\mathsf{Gal}(E/F)$ and intermediate extension fields K with E > K > F.

Fundamental Theorem of Galois Theory

Given a field extension E > F, there is a map $K\mapsto\mathsf{Gal}(E/K)$ taking intermediate extension fields K with E > K > F to subgroups of $\mathsf{Gal}(E/F)$. If E is a splitting field for a polynomial over F, then the map is a one-to-one correspondence.

Some additions to this result follow:
• The index $[E:K]$ of E over K as a vector field equals $|\mathsf{Gal}(E/K)|$ and $[K:F]=|\mathsf{Gal}(E/F)|/|\mathsf{Gal}(E/K)|$.
• If K is the splitting field of a polynomial in $F[x]$ and $\mathsf{Gal}(E/K) \lhd \mathsf{Gal}(E/F)$, then $\mathsf{Gal}(K/F)\cong\mathsf{Gal}(E/F)/\mathsf{Gal}(E/K)$.
• The fixed field of $\mathsf{Gal}(E/K)$ is $E_{\mathsf{Gal}(E/K)}=K$, and for $H<\mathsf{Gal}(E/F)$ we have $H=\mathsf{Gal}(E/E_H)$.

The simplest Galois extensions, those satisfying the requirements in the above theorem, are those adjoining a root of a polynomial. The extension $\mathbb{Q}(\sqrt{2}) > \mathbb{Q}$ can be viewed as $\{a+b\sqrt{2}:a,b\in\mathbb{Q}\}$, and the Galois group $\mathsf{Gal}(\mathbb{Q}(\sqrt{2})/\mathbb{Q})$ is $\mathbb{Z}/2\mathbb{Z}$, with the nonidentity automorphism taking $\sqrt{2}\mapsto -\sqrt{2}$. The fixed field of $\mathsf{Gal}(\mathbb{Q}(\sqrt{2})/\mathbb{Q})$ is $\mathbb{Q}$, as must be the case since any automorphism must fix $\mathbb{Q}$. A second example is the extension $\mathbb{Q}(\omega,\sqrt{2})$ with $\omega=-\frac{1}{2}+i\frac{\sqrt{3}}{2}$ a root of $x^3-1$. The Galois group in this case is actually S3 and thus non-abelian.

A more general example is the extension $GF(p^n) > GF(p) = F$. This can actually be viewed as $F(b)$, where b is the zero of some irreducible degree n polynomial $p(x)\in F[x]$. Any automorphism of $GF(p^n)$ fixes 1 and therefore $GF(p)$ as well. So if $p(b)=0$ then $p(\phi(b))=0$. Thus, there are at most n possibilities for $\phi(b)$. On the other hand, the map $a\mapsto a^p$ is an automorphism of E with order n, since $GF(p^n)^*$ is cyclic. Therefore, $\mathsf{Gal}(GF(p^n)/GF(p))\cong\mathbb{Z}_n$.

# Going Further

## Sylow Theory

Classifying finite abelian groups is relatively easy. It is much more difficult to write down all the nonabelian groups of a certain order. The answer is best achieved with the Sylow theorems, a collection of tools for determining properties of groups of a certain order. We begin by developing some of the theory used in the proofs of the Sylow theorems.

Two subgroups H and K of a group G are conjugate if $H=gKg^{-1}$ for some $g\in G$. For a single element $a\in G$, the conjugacy class $\mathrm{Cl}(a)$ is the set of elements of the form $xax^{-1}$ for $x\in G$. These sets partition the group, but not like cosets since $\mathrm{Cl}(e)=\{e\}$. It is easy to show that $|\mathrm{Cl}(a)|=|G:C(a)|$, where $C(a)$ is the centralizer subgroup. This is used to derive the class equation.

Theorem (Class Equation)

The order of a group G may be written $|G|=\sum |G:C(a)|$, where the sum is taken over conjugacy classes. Alternately, $|G|=|Z(G)|+\sum |G:C(a)|$, where the sum is taken over the nontrivial conjugacy classes. (All elements in $Z(G)$ are in their own conjugacy class.)

This formula reveals a lot about the orders of subgroups. First, if $|G|=p^n$, we can divide the class equation by p to see that $Z(G)$ must be nontrivial. Applying the $G/Z$ Theorem, one finds that all groups of order p2 are abelian. Further consequences follow.

• Sylow's First Theorem: If pk divides |G|, then G has a subgroup of order pk, called a Sylow p-subgroup.
• Sylow's Second Theorem: Every subgroup H < G of order pk is contained in some Sylow p-subgroup.
• Sylow's Third Theorem: The number of Sylow p-subgroups, denoted np, is equal to 1 modulo p and divides |G|. Any two such subgroups are conjugate.

The first theorem is proven inductively using the class equation. The second and third proofs are harder, using the notion of conjugate subgroups and the Orbit-Stabilizer Theorem.

The second theorem implies that a group G has an element of order p for every p dividing |G|. The third theorem implies that a Sylow p-subgroup is unique, meaning $n_p=1$, if and only if it is a normal subgroup.

The above results are extremely useful in classifying the groups of a certain order. As an example, Sylow's Third Theorem implies that a group of order 40 has only one Sylow 5-subgroup; hence that group is normal. A group of order 30 must have either 1 or 6 Sylow 5-subgroups, and 1 or 10 Sylow 3-subgroups. There are not enough elements for both of these subgroups to be non-unique, so one of them must be normal. Therefore, there exists a cyclic, normal subgroup of order 15.

A more general case is $|G|=2p$, for p an odd prime. In this case G is isomorphic to either $\mathbb{Z}/(2p)\mathbb{Z}$ or the dihedral group Dp. In fact, there are at most 2 groups of order pq for any primes p and q. It is also true that for $|G|=pq$, with p < q not dividing $q-1$, then $G\cong\mathbb{Z}/(pq)\mathbb{Z}$.

## Finite Simple Groups

This section continues the theme of understanding and classifying finite groups by looking at the particular case of finite simple groups.

Definition

A simple group is one with no nontrivial normal subgroups. Hence, it has no factor groups.

One can think of simple groups as the building blocks of larger groups. Factoring out the largest normal subgroup G1 of a group G0 gives a simple group $H_1=G_0/G_1$. This process may be continued, and the groups $H_i=G_{i-1}/G_i$ are called the composition factors. Using this decomposition and induction, many problems in group theory can be reduced to the special case of simple groups.

Finite simple groups have been completely classified, divided into seven infinite families and an additional 26 sporadic groups fitting no specific criterion. There are a variety of tests for nonsimplicity. Two of these follow:

• 2*Odd Test: If n = 2m for m odd, then there is no simple group of order n.
• Sylow Test: If p|n and 1 is the only divisor of n that is equivalent to 1 modulo p, then there is no simple group of order n.

The first test is proven using Cayley's Theorem. The second is proven using Sylow's Third Theorem and the fact that prime power groups have nontrivial centers.

Cayley's Theorem on permutations can be generalized to the following: if H < G and S is the group of permutations of left cosets of H in G, then there is a homomorphism $G\to S$ whose kernel is in H and that contains every normal subgroup of G that is also in H. This result gives two more tests:

• Index Test: If G is a finite group and H < G with |G| not dividing $|G:H|!$, then H contains a nontrivial normal subgroup and is not simple.
• Embedding Test: If a finite non-abelian group G with a subgroup of index n is not isomorphic to a subgroup of An, then it is not simple.

A simple application of these four tests shows that the only possible orders less than 200 for a non-abelian simple group are 60, 72, 112, 120, 144, 168, and 180. Of these, only 60 are 168 actually correspond to simple groups. The others may be eliminated with some additional work. In the case of 72, Sylow's Third Theorem implies that there are either 1 or 4 Sylow 3-subgroups. The first case would imply a normal subgroup, so $n_3=4=|G:N(H)|$. But then G cannot be simple by the Index Test, since |G| does not divide 4!. The order 60 simple group is A5, which may be proven to be simple by eliminating also possible orders for subgroups.

Further study of abstract algebra involves several additional algebraic structures, beyond groups, rings, and fields. The most useful is probably the module, which is a generalization of a vector space, this time over a ring. It also generalizes the notion of a group acting on a set (like a permutation group). All groups are modules, and many theorems for groups (like the classification of finite abelian ones) carry over to modules. Another structure is the algebra, which one can think of as a ring with a little less structure. Other combinations of axioms may lead to other kinds of structures.

Representation theory analyzes a group using its homomorphisms, and particularly by its homomorphisms into particular matrix groups. In this case, the properties of the traces of the map's image provide a great deal of information about the group itself. Lie groups and Lie algebras are another key idea that are often tied directly to representation theory. A Lie group is a group with a differential structure, or equivalently a manifold with a group structure. A Lie algebra is an algebra with an anti-commutative product $[X,Y]=-[Y,X]$ that satisfies a particular identity. Lie algebras can be completely classified, and, amazingly, there is a close correspondence between Lie groups and Lie algebras; with a few minor conditions on the Lie group, there is in fact a 1:1 correspondence. Lie theory lies at the intersection of geometry, topology, and algebra, which makes it useful to a huge range of mathematicians.

Finally, category theory attempts to encapsulate the commonalities between various structures. Often, theorems can be phrased in terms of category theory so as to generalize theorems that exist separately in different contexts. The basic objects in category theory are objects and morphisms, which are special functions between the objects. Examples here include groups and homomorphisms between groups, or rings and ring homomorphisms between rings. There are lots of additional examples; category theory is general enough to encompass many diverse mathematical structures.

# References

The following are two recommended texts frequently used for undergraduate courses in abstract algebra.

1. John B. Fraleigh, "A First Course in Abstract Algebra", Addison Wesley, 2002.
2. Joseph A. Gallian, "Contemporary Abstract Algebra", Brooks Cole, 2004.
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