114

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Course Name:	114 Abstract Algebra
Lecturer:	Dr. Donal O'Donovan
Lecturer's Page:	http://www.maths.tcd.ie/~don/
Course Page:	http://www.maths.tcd.ie/pub/official/Courses07-08/114.html

Notes on course 111 (abstract + linear algebra), 2006-2007 are here

A short page on some theorems that come up in Donal O'Donovan's exams is here

A page that gives some examples of the definitions below can be found here

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Mappings and Functions

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Definitions and Properties

A mapping or a function is a relation that assigns to each value of a set, say $S$ , a unique element of another set, call it $T$ . $S$ is called the domain, and $T$ the co-domain. This mapping is often written $f:S\rightarrow T$ .

Two mappings, $f$ and $g$ are said to be equal if:

The domains are equal
The codomains are equal
$f(x)=g(x)$ for all $x$ in the common domain.

A mapping $f : S\rightarrow T$ is said to be one-to-one or an injective mapping if $f(x_1) = f(x_2) \Rightarrow x_1 = x_2$ for all $x_1, x_2 \in S$ . Namely, every point in $S$ gets mapped to at most one point in $T$ .

A mapping $f :S\rightarrow T$ is said to be onto or a surjective mapping if for all $y\in T$ , there exists some $x\in S$ such that $f(x)=y$ . Namely, every point in $T$ has at least one point mapped to it from $S$ .

A mapping that is both injective and surjective (namely one-to-one and onto) is called bijective.

The set of all mappings from a set $S$ to itself is denoted $M(S)$ . The set of all bijective mappings from $S$ to $S$ is denoted $\mathrm{Sym}(S)$ .

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Composition and Inversion

Given two mappings $f : S \rightarrow T$ and $g : T \rightarrow U$ , the composition of the mappings is defined as $(g \circ f)(x) = g(f(x))$ for all $x\in S$ .

Theorem: Given two mappings $f : S \rightarrow T$ and $g : T \rightarrow U$ , the following are true:

$f$ and $g$ are both onto mappings imply that $g \circ f$ is an onto mapping,
$g\circ f$ is an onto mapping implies that $g$ is an onto mapping,
$f$ and $g$ are both one-to-one mappings imply that $g\circ f$ is a one-to-one mapping
$g\circ f$ is a one-to-one mapping implies that $f$ is a one-to-one mapping.

The Identity Mapping on a set $S$ is defined as $\iota_S(x)=x$ for all $x \in S$ .

Inverses: Given two mappings $f : S \rightarrow T$ and $g : T \rightarrow S$ , we say that $g$ is a right inverse of $f$ if $(f \circ g)(y) = \iota_T(y)$ for all $y \in T$ . Similarly, we say that $g$ is a left inverse of $f$ if $(g \circ f)(x) = \iota_S(x)$ for all $x \in S$ .

Lemma: Left and right inverses of any mapping are equal, and inverse mappings are unique.

Lemma: A mapping is invertible iff it is bijective.

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Operations

A binary operation $*$ on a set $S$ is a relationship that assigns to each ordered pair of $S$ a unique element of $S$ . By definition, if $a\in S$ and $b\in S$ then $a* b \in S$ . This is referred to as closure of the operation $*$ on the set $S$ .

A Cayley table is a grid of operations of the elements of a set $S$ . Consider the set $S = \left \lbrace a,b,c \right\rbrace$ . The Cayley table for $S$ under some operation $*$ is:

$*$	$a$	$b$	$c$
$a$	$a*a$	$a*b$	$a*c$
$b$	$b*a$	$b*b$	$b*c$
$c$	$c*a$	$c*b$	$c*c$

An operation $*$ on a set $S$ is said to be associative if $a* (b* c)= (a* b)* c$ for all $a,b,c\in S$ .

An operation $*$ on a set $S$ is said to be commutative if $a* b= b* a$ for all $a,b\in S$ .

For an operation $*$ on a set $S$ , if there exists an element $e\in S$ such that $a* e= e* a = a$ for all $a\in S$ , then $e$ is said to be an Identity Element of $S$ .

For an operation $*$ on a set $S$ , given some $a\in S$ , if there exists an element $b\in S$ such that $a* b= b* a = e$ , then $b$ is said to be an inverse of $a$ . The inverse of the element $a$ is more often written $a^{-1}$ .

Lemma: If an identity element exists for an operation, it is unique.

Lemma: If an inverse element exists for an element under an operation, it is unique.

Lemma: Composition is an associative operation on the set of all mappings, $M(S)$ .

Lemma: Composition is an operation on $\mathrm{Sym}(S)$ .

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Groups and Permutations

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Introduction to Groups

A set $G$ and an operation $*$ is said to be a group, denoted $(G,*)$ , if the following properties hold.

Associativity: $a* (b* c)= (a* b)* c$ for all $a,b,c\in S$
Existence of an Identity Element: there exists an $e \in G$ such that for all $a\in G$ , $a* e= e* a = a$
Existence of Inverses: for all $a\in G$ there exists a $a^{-1} \in G$ sich that $a* a^{-1}= a^{-1}* a = e$

Lemma: For a group $(G, *)$ , the identity element is unique and inverses are also unique.

A set $G$ and an operation $*$ on that set is said to be a semi-group if the operation associative on $G$ .

A group $(G, *)$ is said to be an Abelian Group if $*$ is commutative, i.e. if $a* b= b* a$ for all $a,b\in G$ .

The order of a group $(G, *)$ is defined to be the amount of elements in the set, and is denotes as $|G|$ .

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Permutations

A permutation is defined as a one-to-one mapping from a set $S$ onto itself, such that the elements form a group. It is usually denoted by:

$\sigma = \begin{pmatrix} 1 & 2 & 3 & \dots & n \\ p_1 & p_2 & p_3 & \dots & p_n \end{pmatrix}$ where $p_i \in \left\lbrace 1,2,3,\dots ,n \right\rbrace$ . Note that the top line need not always be consecutive natural numbers.

Seeing as a permutation is bijective, the set of all permutation on a set $S$ is the same as $\mathrm{Sym}(S)$ . Note that the operation on permutations is always composition.

$(\mathrm{Sym}(S), *)$ is a group for any set $S$ .

If $S= \left\lbrace1,2,\dots ,n \right\rbrace$ then $\mathrm{Sym}(S)$ is more commonly denoted as $S_n$ .

Lemma: $|S_n|= n!$ .

All permutations may be reduced from the two-level notation to the product of single-line permutations representing the individual cycles within the given permutation, e.g.

$\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6\\ 2 & 1 & 3 & 5 & 5 & 4 \end{pmatrix} = \begin{pmatrix}1 & 2 \end{pmatrix} \begin{pmatrix} 3 \end{pmatrix} \begin{pmatrix} 4 & 5 & 6 \end{pmatrix}$ . We may remove the identity permutation on $(3)$ , as it is implied if not explicitly stated. The above is called a $k$ -cycle, or a product of a 2-cycles and a 3-cycle.

For any cycles $a = \begin{pmatrix} a_1 & a_2 & \dots & a_m \end{pmatrix}$ and $b = \begin{pmatrix} b_1 & b_2 & \dots & b_n \end{pmatrix}$ , $a$ and $b$ are said to be disjoint if $a_i \neq b_j$ for any $1 \leq i \leq m$ , $1 \leq j \leq n$ .

Lemma: Disjoint cycles commute.

Lemma: If $X$ is a finite set, and $\sigma \in \mathrm{Sym}(X)$ then $\sigma$ is a product of disjoint cycles.

Lemma: Every permutation can be written as a product of 2-cycles.

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Subgroups

A subset $H$ of a group $(G, *)$ is a subgroup of $G$ if it is a group itself with respect to the inherited operation. This is written $H < G$ .

Theorem: If $H\subseteq G$ then $H<G$ iff $H$ has the following properties:

$H \neq \emptyset$
Closure: $a\in H$ and $b\in H$ imply that $a* b \in H$
Existence of Inverses: for all $a\in H$ there exists some $a^{-1} \in H$ such that $a* a^{-1}= a^{-1}* a = e$ .

Lemma: If $H$ is a subgroup of $G$ , then

The identity element in $H$ is the identity element in $G$ , and
The inverse of the element $a\in H$ is the same as the inverse of $a$ in $G$ .

Lemma: A transposition is any 2-cycle in $S_n$ .

A Permutation is said to be an odd or an even permutation if it can be decomposed to an odd or even number of 2-cycles respectively.

Lemma: The set of all even permutations in $S_n$ form a subgroup of $S_n$ for all $n\geq 2$ . This subgroup is called the Alternating Group, denoted by $A_n$ .

Lemma: $|A_n| = \frac{1}{2} n!$

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Symmetry

Let the set $P$ denote all the points in a plane, and $M$ denote the set of all permutations that preserve the distance between the points in $P$ . The permutations in $M$ are said to be isometries, or motions, of the plane.

For any fixed point $p\in P$ , any movement of the plane about the point $p$ that remains a fixed distance from $p$ is called a rotation, and is a motion of the plane.

For any fixed point $p\in P$ and line $L\in P$ , the mapping that sends the point $p$ to $q$ such that $L$ is the perpendicular bisector of the line joining $p$ and $q$ is called a reflection of the plane through $L$ , and is a motion of the plane.

For any fixed points $p,q\in P$ , a translation is the mapping that sends the points $p$ and $q$ a fixed distance in the same direction, and is a motion of the plane.

Lemma: $M$ , the set of all isometries of the plane, is a subgroup of $\mathrm{Sym}(P)$ .

For any set of points on a plane, denoted by $T$ , the group of all motions of that remain within $T$ is called the group of symmetries of $T$ and is denoted by $M_{(T)}$ .

An n-gon is a shape with $n$ sides, all of which are equal in length, whose internal angles are all equal in size, e.g. the 3-gon is an equilateral triangle, the 4-gon is a square, etc.

For any n-gon, we denote $D_n$ to be all the symmetries of that n-gon. These symmetries form a group under composition.

$|D_n| = 2n$

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Equivalence and Congruence

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Equivalence Relations

A relation $\sim$ is said to be an equivalence relation on a non-empty set $S$ if the following properties hold:

Reflexive: for all $a\in S$ , $a\sim a$
Symmetric: for all $a,b \in S$ , $a\sim b \Rightarrow b\sim a$
Transitive: for all $a,b,s\in S$ , if $a\sim b$ and $b\sim c$ , then $a\sim c$

A collection of non-empty subsets of a non-empty set $S$ forms a partition of $S$ if:

The union of all subsets of the partition is $S$
If $A$ and $B$ are subsets of the partition, then they are either equal or disjoint, i.e. $A=B$ or $A \cap B = \emptyset$

The equivalence class of an element $a$ in a non-empty set $S$ is defined as $\left[ a \right]= \left\lbrace x : x\sim a \right\rbrace$

The equivalence class representatives of a relation $\sim$ is said to be the set containing precisely one element from every equivalence class of $\sim$ .

Lemma: Let $G$ be a permutation group on a set $S$ and define the relation $\sim$ by $a\sim b \Leftrightarrow f(a)=b$ where $f \in G$ . Then $\sim$ is an equivalence relation on $S$ .

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Congruence

For a natural number $n$ , integers $a$ and $b$ are said to be congruent modulo n if $n$ divides $a-b$ . This is written $a\equiv b \mod n$ .

The relation $a\sim b$ defined as $a\equiv b \mod n$ is an equivalence relation for integers $a$ and $b$ and natural number $n$ .

The Least Integer Principle: Every non-empty set of positive integers contains a least element, namely, the minimum.

The Division Algorithm: If $a$ and $b$ are integers with $b>0$ , then there exist unique integers $q$ and $r$ such that $a = bq+r$ where $0\leq r < b$ .

Lemma: Letting $n$ be a positive integer, every integer is congruent modulo $n$ to exactly one of $\left\lbrace 0, 1, 2, \dots, n-1 \right\rbrace$ .

Given the above equivalence relation of congruence, let $\left[ k \right]$ denote the equivalence class to which $k$ belongs, modulo $n$ . This is called the congruence class of $k$ modulo $n$ .

Let $\mathbb{Z}_n$ denote the set of all equivalence classes for the above relation, namely $\mathbb{Z}_n = \left\lbrace \left[ 0 \right], \left[ 1 \right], \dots, \left[ n-1 \right] \right\rbrace$

There are two main operations that will be used on this set of congruence classes. Define $\left[ a \right] \oplus \left[ b \right] = \left[ a+b \right]$ and $\left[ a \right] \otimes \left[ b \right] = \left[ ab \right]$ . It can be proven that these operations are well defined. Sometimes, regular addition and multiplication signs are used for the addition/multiplication of congruence classes modulo $n$ where the meaning is clear.

Theorem: $\mathbb{Z}_n$ is an Abelian group with respect to $\oplus$ .

Lemma: There is a group of order $n$ for every positive integer $n$ .

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Divisors and Multiples

Theorem: Given any two integers $a$ and $b$ , not both equal to zero, then there exists a unique positive integer $d$ , called the greatest common divisor and denoted $\left( a,b \right)$ or sometimes $\mathrm{gcd}(a,b)$ , such that

$d\mid a$ and $d\mid b$ , and
if $c$ is an integer such that $c\mid a$ and $c\mid b$ , then $c\mid d$

Theorem: For all integers $a$ and $b$ , not both equal to zero, there exists integers $m$ and $n$ such that $\left( a,b \right) = am+bn$

If $\left( a,b \right) = 1$ , we say that $a$ and $b$ are relatively prime, or coprime.

Theorem: Given any two integers $a$ and $b$ , not both equal to zero, then there exists a unique positive integer $m$ , called the lowest common multiple and denoted $\left[ a,b \right]$ , such that

$a\mid m$ and $b\mid m$ , and
if $c$ is an integer such that $a\mid c$ and $b\mid c$ , then $m\mid c$

Lemma: For all integers $a,b,c$ , if $a\mid bc$ and $\left( a,b \right) = 1$ then $a\mid c$ .

Corollary: If $p$ is a prime number, and $a_1, a_2, \dots, a_n$ are integers such that $p\mid a_1 a_2 \dots a_n$ , then $p\mid a_i$ for some $0\leq i \leq n$ .

The Fundamental Theorem of Arithmetic: Each integer greater than 1 can be written as a product of powers of prime numbers and, except for the order in which this is done, this can only be done in one way.

For all integers $n>1$ , let $\phi (n)$ denote the number of positive integers that are less than $n$ and relatively prime to $n$ . This function is known as Euler's Phi Function. Note that $\phi (1)=1$ .

Lemma: Given any prime number $p$ and positive integer $r$ , $\phi (p^r) = p^r - p^{r-1}$ .

Lemma: If $m$ and $n$ are two integers such that $\left( m,n \right) = 1$ , then $\phi (mn) = \phi (m) \phi(n)$ .

Lemma: Given any integer $n$ , which can be written as a product of the powers of prime numbers, say $n = p_1 ^{e_1} p_2 ^{e_2} \dots p_k ^{e_k}$ , then $\phi (n) = n \left( 1 - \frac{1}{p_1} \right) \left( 1 - \frac{1}{p_2} \right) \dots \left( 1 - \frac{1}{p_k} \right)$

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Groups Revisited

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Notation and Cyclic Groups

Notation: Note that from now on, to avoid the awkward $*$ notation, we will use the multiplication notation instead, purely for convenience. So, $a* b$ will now be just $ab$ . Similarly, $a* a$ will be written $a^2$ , $a* a* a$ will be $a^3$ , etc. If the operation is explicitly defined, then the appropriate notation should be used. For example, when the operation is addition, $a* b$ becomes $a+b$ , $a* a$ becomes $2a$ , etc.

Theorem: Let $G$ be a group.

If $a,b,c\in G$ , and $ab=ac$ or $ba=ca$ , then $b=c$ .
If $a, b\in G$ , then the equations $ax=b$ and $xa=b$ have unique solutions in $G$ .
If $a\in G$ , then $(a^{-1})^{-1} = a$ .
If $a,b \in G$ , then $(ab)^{-1} = b^{-1}a^{-1}$ .

Lemma: Let $G$ be a group. For $a\in G$ , the set of all integral powers of $a$ , namely the set defined as $\left\langle a\right\rangle = \left \lbrace a^n : n\in\mathbb{Z} \right \rbrace$ is a subgroup of $G$ .

If $G$ is a group such that $G = \left\langle a\right\rangle$ for some $a\in G$ , then we say that $G$ is a cyclic group.

Lemma: If $G$ is a cyclic group and $H<G$ , then $H$ is also a cyclic group.

Theorem: Let $G$ be a group with $a\in G$ , and let there exist unequal integers $r$ and $s$ such that $a^r = a^s$ .

There exists a smallest positive integer $n$ such that $a^n=e$ ,
If $t$ is an integer, then $a^t = e$ iff $n|t$ , and
The elements $e, a, a^2, \dots, a^{n-1}$ are distinct, and $\left\langle a \right\rangle = \left\lbrace e, a, a^2, \dots, a^{n-1} \right\rbrace$ .

Lemma: If a group $G$ is cyclic, then it is also Abelian.

If $a$ is an element of a group, then the smallest positive integer $n$ such that $a^n=e$ , should one exist, is called the order of $a$ , denoted $o(a)$ . If $n$ doesn't exist, $a$ is said to be of infinite order.

Lemma: If a is an element of a group, then $o(a) = | \left\langle a \right\rangle |$ .

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Generators

Lemma: Let the groups $H_1, H_2, \dots, H_k$ be subgroups of a group $G$ . Thus, $H_1 \cap H_2 \cap \dots \cap H_k$ is also a subgroup of $G$ .

Theorem: Let $S$ be a subset of $G$ , and let $\left\langle S \right\rangle$ be the intersection of all the subgroups of $G$ that contains $S$ . Thus, $\left\langle S \right\rangle$ is the unique smallest subgroup of $G$ that contains $S$ . This means that:

$S \subseteq \left\langle S\right\rangle$
$\left\langle S\right\rangle < G$
If $H < G$ with $S \subseteq H$ , then $\left\langle S\right\rangle \subseteq H$

Note: we say that $\left\langle S\right\rangle$ is generated by the set $S$ , and that $S$ generates the set $\left\langle S\right\rangle$ .

Before, we considered the set $\left\langle a\right\rangle$ for an elememt $a$ in a group. However, we can extend this to more than one element of the group. Consider the set $S = \left\lbrace a_1, a_2, \dots, a_n\right\rbrace$ . Let $S = \left\langle a_1, a_2, \dots, a_n\right\rangle$ denote the set of integral powers and combinations of all the elements $a_1, a_2, \dots, a_n$ .

Lemma: Let $T_1$ and $T_2$ be subsets of a group $G$ . Thus $\left\langle T_1\right\rangle = \left\langle T_2\right\rangle$ iff $T_1 \subseteq \left\langle T_2\right\rangle$ and $T_2 \subseteq \left\langle T_1\right\rangle$ .

Given any two sets $A$ and $B$ , we define a new set $A\times B$ called the cross product, or Cartesian product, of $A$ and $B$ . This set is defined as $A\times B = \left \lbrace (a,b): a\in A, b\in B \right \rbrace$ .

Lemma: Given any two groups $A$ and $B$ , the cross product $A\times B$ is also a group, with respect to the inherited operations. The operation is given by $(a_1, b_1)(a_2, b_2) = (a_1a_2, b_1b_2)$ for all $a_1,a_2 \in A$ and $b_1, b_2 \in B$ .

Lemma: If $A$ and $B$ are finite groups, then $|A\times B| = |A|\cdot |B|$ .

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Rings, Integral Domains and Fields

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Rings

A Ring is a set $R$ with two operations, addition and multiplication, such that the following properties hold:

$R$ is an Abelian group under addition
Multiplication is associative
Left and right distributive laws hold for all values in $R$

We shall use the generic $+$ and $\cdot$ to represent addition and multiplication respectively. The identity element for the operation addition will be called the zero element of $R$ and denoted $0$ .

Lemma: $\mathbb{Z}_n$ is a ring under $\oplus$ and $\otimes$ .

Lemma: Given two rings $R$ and $S$ , the direct product of the rings, namely $R\times S$ is also a ring.

Theorem: Let $R$ be a ring, and let $a,b, c \in R$ .

The zero element in $R$ is unique,
Every element has a unique inverse,
If $a+b=a+c$ or $b+a=c+a$ , then $b=c$ ,
Every equation of the form $x + a = b$ or $a+x = b$ has a unique solution in $R$ ,
$-(-a) = a$ and $-(a+b)= (-a)+(-b)$ ,
If $m$ and $n$ are integers, then $(m+n)a = ma+na$ , $m(a+b) = ma+mb$ and $m(na) = (mn)a$ ,
$0\cdot a = a\cdot 0 = 0$ ,
$a(-b) = (-a)b = -(ab)$ ,
$a(b-c) = ab-ac$ and $(a-b)c = ac-bc$ .

Note that for a ring, nothing but associativity is assumed for multiplication. If multiplication happens to be commutative, we say that the ring is an Abelian Ring. If there exists an identity element for multiplication, it is simply called the identity and denoted $e$ .

Let $R$ be a commutative ring, and let $a\in R$ with $a\neq 0$ . If there exists a $b\in R$ with $b\neq 0$ such that $ab=0$ , then $a$ and $b$ are said to be zero divisors of $R$ .

A subset $S$ of a ring $R$ is a subring of $R$ if it is a ring itself with respect to the inherited operations. This is written $S < R$ .

Let $R$ be a ring. If $S\subseteq R$ then $S < R$ iff S has the following properties:

$S \neq \emptyset$ ,
Closure: $a\in S$ and $b\in S$ imply that $a+b \in S$ and $ab\in S$ , and
Existence of Inverses: for all $a\in S$ there exists some $-a \in S$ such that $a+(-a)= -a + a = e$ .

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Integral Domains and Fields

An integral domain is a commutative Ring with and identity $e\neq 0$ and with no zero divisors.

Lemma: Given an integral domain $D$ , with $a,b,c\in D$ and $a\neq 0$ , then $ab=ac$ implies that $b=c$ .

A commutative ring in which the set of non-zero elements form a group with respect to multiplication is called a field. Alternatively, a field is an integral domain in which every non-zero element has a multiplicative inverse.

Lemma: Every finite integral domain is a field

Theorem: \mathbb{Z}_n is a field iff n is a prime number.

A subset $K$ of a field $F$ is a subfield of $F$ if it is a field itself with respect to the inherited operations. This is written $K < F$ .

Lemma: Let $F$ be a field. If $K\subseteq F$ then $K < F$ iff $K$ has the following properties:

$K \neq \emptyset$
Closure: $a\in K$ and $b\in K$ imply that $a+b \in K$ and $ab\in K$ , and
Existence of Inverses: if $a\in K$ , then $-a\in K$ and if $a\neq 0$ then $a^{-1}\in K$ .

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More on Rings

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Isomorphism

Let $R$ and $S$ be rings. An isomorphism of $R$ onto $S$ is a bijective mapping $\theta : R \rightarrow S$ such that $\theta (a+b) = \theta (a) + \theta (b)$ and $\theta (ab) = \theta (a)\cdot\theta (b)$ for all $a,b\in R$ . Similarly with group isomorphisms, we write $R\cong S$ .

Let $R$ be a ring. If there exists a positive integer $n$ such that $na = 0$ for all $a\in R$ , then the least such integer is called the characteristic of $R$ . If such an integer does not exist, then $R$ is said to have characteristic 0.

Lemma: If $D$ is an integral domain, then $D$ either has characteristic 0, or a prime characteristic.

Lemma: Let $D$ be an integral domain. If $D$ has characteristic 0, then $D \cong \mathbb{Z}$ . If D has a characteristic $p$ , then $D\cong\mathbb{Z}_p$ .

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Homomorphisms

If $R$ and $S$ are rings, a mapping $\theta : R\rightarrow S$ is a ring homomorphism if, for all $a,b\in R$ , $\theta(a+b) = \theta(a)+\theta(b)$ and $\theta(ab) = \theta(a)\theta(b)$ .

If $\theta :R\rightarrow S$ is a ring homomorphism, then the kernel of $\theta$ is denoted $\ker\theta$ and is defined as the set of all elements $r\in R$ such that $\theta(r) = 0$ , where $0$ is the zero in S.

A subring $I$ of a ring $R$ is called an ideal of $R$ if $ar\in I$ and $ra\in I$ for all $a\in I$ and $r\in R$ .

Theorem: Let $\theta:R\rightarrow S$ be a ring homomorphism.

The image of $\theta$ is a subring of $S$ ,
The kernel of $\theta$ is an ideal of $R$ , and
$\theta$ is one to one iff $\ker\theta = {0_R}$ , where $0_R$ is the zero for $R$ .

Let $R$ be a commutative ring with identity $e$ , and let $a\in R$ . We denote the set of all multiples of $a$ by $\left\langle a\right\rangle =\left \lbrace ra\;:\;r\in R \right \rbrace$ . It can be proved that this set is an ideal of $R$ , and we call it a principal ideal.

Theorem: Let $I$ be an ideal of a ring $R$ , and let $R/I$ denote the set of all right cosets of $I$ considered as a subgroup of the additive group of $R$ . For $I+a, I+b \in R/I$ , define the operations $(I+a)+(I+b) = (I+(a+b))$ and $(I+a)(I+b)=(I+ab)$ . With these operations, $R/I$ is a ring, called the quotient ring of $R$ .

Lemma: If $I$ is an ideal of the ring $R$ , then the mapping $\eta : R\rightarrow R/I$ defined by $\eta(a) = I+a$ for all $a\in R$ is a homomorphism of $R$ onto $R/I$ , and $\ker\eta = I$ .

Fundamental Theorem Of Homomorphisms: Let $R$ and $S$ be rings and let $\theta : R\rightarrow S$ be a homomorphism from $R$ onto $S$ with $\ker\theta = I$ . The mapping $\phi : R/I\rightarrow S$ defined by $\phi(I+a) = \theta(a)$ is an isomorphism of $R/I$ onto $S$ , and thus $R/I\cong S$ .

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Constructing Fields

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The Integers

Consider the set of positive whole numbers, $\mathbb{N} = \left \lbrace 1, 2, 3, \dots \right \rbrace$ . This set of numbers has no solution for $1+x = 0$ , for example. More generally, $a+x = b$ does not always have a solution in $\mathbb{N}$ . Let $(a,b)$ denote the solution of the equation $a+x = b$ . As such equations may have more than one solution, define the relation below.

Define $(a,b) \sim (c,d)$ means $a+d = b+c$ . This is an equivalence relation (check). Thus, the set of integers is defined as the set of equivalence classes for this relation, namely $\mathbb{Z} = \mathbb{N}\times\mathbb{N}/\sim.$

Lemma: $\mathbb{Z}$ is a ring with respect to normal addition $(\oplus)$ and multiplication $(\otimes)$ . For simplicity, these symbols may be omitted.

An element $a$ in a subset $S$ of an ordered integral domain $D$ is a least element of $S$ if $x > a$ for all $x\in S$ such that $x\neq a$ .

An ordered integral domain is said to be well ordered if every non-empty subset of $D^p$ has a least element.

Lemma: If $D$ is a well ordered integral domain, with identity $e$ , then $e$ is the least element of $D^p$ .

Theorem: If $D$ is a well ordered integral domain, then $D$ is isomorphic to the ring of integers.

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The Rational Numbers

In parallel with the previous section, consider the set of integers, $\mathbb{Z} = \left \lbrace \dots, -2, -1, 0 1, 2, \dots \right \rbrace$ . This set of numbers has no solution for $2x = 1$ , for example. More generally, $ax = b$ does not always have a solution in $\mathbb{Z}$ . Let $(b,a)$ denote the solution of the equation $ax = b$ , where $a\neq 0$ . As such equations may have more than one solution, define the relation below.

Define $(b,a) \sim (d,c)$ means $ad = bc$ . This is an equivalence relation (check). Thus, the set of rationals is defined as the set of equivalence classes for this relation, $namely \mathbb{Q} = \mathbb{Z}\times\mathbb{Z}^\prime/\sim$ . Note that $\mathbb{Z}^\prime = \mathbb{Z} \setminus {0}$ . On this set, define the following relations:

$[(a, b)] + [(c,d)] = [(ad+bc, bd)] \quad [(a,b)]\cdot [(c,d)] = [(ac,bd)]$ These operations are well defined (check).

Lemma: Given the set of rational numbers $\mathbb{Q}$ , consider $\mathbb{Q}^p = \left \lbrace x\;:\;x\in\mathbb{Q}, x>0 \right \rbrace$ . Thus, $\mathbb{Q}$ is an ordered field, with $\mathbb{Q}^p$ as its positive elements. Note that an ordered field is simply an ordered integral domain that is also a field.

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The Real Numbers

An element $u$ of an ordered field $F$ is an upper bound of a subset $S$ of $F$ if $u\geq X$ for all $x\in S$ . An element $u\in F$ said to be the least upper bound for a subset $S$ or $F$ if:

$u$ is an upper bound for $S$ , and
if $v\in F$ is an upper bound for $S$ , then $v\geq u$ .

An ordered field $F$ is said to be complete if every non-empty subset of $F$ having an upper bound in $F$ has a least upper bound in $F$ .

Theorem: There exists a complete ordered field. Any two such fields are isomorphic, and any such field contains a subfield isomorphic to the field of rationals.

The field of real numbers \mathbb{R} is such a field. The actual construction of this field, however, is somewhat complicated, and is not required, so will be omitted.

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The Complex Numbers

Theorem: Let $\mathbb{C} = \left \lbrace (a,b)\;:\;a,b\in\mathbb{R} \right \rbrace$ , and define addition and multiplication of these ordered pairs for all a,b,c,d\in\mathbb{R} as:

$(a,b)+(c,d)=(a+c,c+d)\quad (a,b)(c,d)=(ac-bd, ad+bc)$ With these operations, $\mathbb{C}$ is a field, called the field of complex numbers.

Lemma: The subset of $\mathbb{C}$ consisting of all $(a,0)$ with $a\in\mathbb{R}$ is a subfield of $\mathbb{C}$ and is isomorphic to $\mathbb{R}$ .

The Fundamental Theorem of Algebra: For all $a_0, a_1, \dots , a_n\in\mathbb{C}$ , the expression $a_0+a_1x+a_2x^2+\dots +a_nx^n$ has at least one solution in the field of complex numbers.

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Others

A number $\alpha$ is said to be a algebraic over a field $F$ if there exists an equation $a_0+a_1x+a_2x^2+\dots+a_nx^n$ with $a_0,a_1,\dots,a_n\in F$ such that $a_0+a_1\alpha+a_2\alpha^2+\dots+a_n\alpha^n = 0$ . The set of variables that satisfy this property over the field $\mathbb{C}$ is called the set of algebraic numbers, denoted $\mathbb{A}$ .

The complement of $\mathbb{A}$ , i.e. the set of all numbers that are not algebraic over $\mathbb{C}$ , is called the set transcendental numbers, denoted $\mathbb{T}$

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Polynomials

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Introduction

Given a commutative ring $R$ , a polynomial in indeterminate $x$ is given by $p(x) = a_0+a_1x+a_2x^2+\dots+a_nx^n$ , where $a_0,a_1,\dots,a_n\in R$ .

Given a polynomial in indeterminate $x$ over a field $R$ , we define the following terms:

Coefficient: any of the values $a_0,a_1,\dots,a_n\in R$ that determine the polynomial.
Leading Coefficient: the non-zero coefficient of the highest power of $x$ .
A Monic Polynomial: a polynomial whose leading coefficient is 1.
The Degree of a Polynomial: the power of $x$ for the leading coefficient.
$R[x]$ : the set of all polynomials over the field $R$ .

Let $p(x) = a_0+a_1x+a_2x^2+\dots+a_mx^m$ and $q(x) = b_0+b_1x+b_2x^2+\dots+b_nx^n$ be polynomials over a commutative ring $R$ . Without loss of generality, assume that $m\geq n$ , the proof for $n > m$ being similar. Thus, we define the following:

$p(x) + q(x) = (a_0+b_0)+(a_1+b_1)x+\dots +(a_n+b_n)x^n + a_{n+1}x^{n+1}+\dots+a_mx^m$ $p(x)q(x) = c_0+c_1x+\dots+c_{m+n}x^{m+n}$ Where $c_k = a_0b_k+a_1b_{k-1}+\dots+a_kb_0$

Lemma: Given any commutative ring $R$ , the set $R[x]$ is a commutative ring with respect to the above operations. If $R$ is an integral domain, then $R[x]$ is also an integral domain.

Theorem: Given a field $F$ , if $f,g \in F[x]$ and $g(x) \neq 0$ , then there exist unique polynomials $q,r\in F[x]$ such that $f(x) = g(x)q(x)+r(x)$ , where $r(x) = 0$ or $\deg r(x) < \deg f(x)$ .

The Remainder Theorem: Given a field $F$ , with $f\in F[x]$ and $c\in F$ , the remainder of $f(x)$ when divided by $(x-c)$ is $f(c)$ .

The Factor Theorem: Given a field $F$ , with $f\in F[x]$ and $c\in F$ , then $c$ is a factor of $f(x)$ iff $f(c) = 0$ .

Theorem: Given a field $F$ with $a,b\in F[x]$ , where $a$ and $b$ are not both zero, there exists a unique monic polynomial $d(x)$ such that:

$d(x)\mid a(x)$ and $d(x)\mid b(x)$ , and
if $c\in F[x]$ is a polynomial such that $c(x)\mid a(x)$ and $c(x)\mid b(x)$ , then $c(x)\mid d(x)$

This polynomial $d$ is called the greatest common divisor of $a$ and $b$ . Note that even the proof of this theorem is a direct parallel to the theorem of the greatest common divisor of integers.

Lemma: Given a field $F$ with $a,b\in F[x]$ , where $a$ and $b$ are not both zero, where $d\in F[x]$ is the greatest common divisor of $a$ and $b$ , there exist unique polynomials $u,v\in F[x]$ such that $d(x) = a(x)u(x)+b(x)v(x)$ .

Two polynomials $f(x)$ and $g(x)$ over a field $F$ are said to be associates if there exists a $c\in F$ such that $f(x) = c\cdot g(x)$ . If a polynomial $f$ has no divisors other that its associates, and is of degree at least 1, then it is called irreducible.

Lemma: If $F$ is a field and $a,b,p\in F[x]$ , where $p$ is irreducible and $p(x)\mid a(x)b(x)$ , then $p(x) \mid a(x)$ or $p(x)\mid b(x)$ .

Corollary: If $F$ is a field and $p,a_1,a_2,\dots, a_n\in F[x]$ , where $p$ is irreducible and $p(x)\mid a_1(x)a_2(x)\cdots a_n(x)$ , then $p(x) \mid a_i(x)$ for some $1\leq i\leq n$ .

Theorem: Every polynomial of degree at least 1 over a field $F$ cam be written as an element of $F$ times a product of monic, irreducible polynomials in $F[x]$ and, except for the order in which the monic polynomials can be written, this can only be done in one way.

Theorem: Let $F$ be a field, and let $p\in F[x]$ . $F[x]/\left\langle p(x) \right\rangle$ is a field iff $p(x)$ is irreducible over $F$ .

Lemma: Let $F$ be a field, $p(x) = a_0+a_1x+a_2x^2+\dots+a_nx^n$ be a polynomial over $F$ , and let $I$ be the ideal $\left\langle p(x) \right\rangle$ . Each element of $F[x]/I$ can be expressed uniquely in the form $I+(b_0+b_1x+b_2x^2+\dots+b_{n-1}x^{n-1})$

Theorem: The ring $\mathbb{R}/\left\langle x^2+1 \right\rangle$ is a field, and also $\mathbb{R}/\left\langle x^2+1 \right\rangle \cong \mathbb{C}$ .

Lemma: If $F$ is a field, then ever ideal of the polynomial ring $F[x]$ is a principal ideal.

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From Mathsoc wiki

Contents

Mappings and Functions

Definitions and Properties

Composition and Inversion

Operations

Groups and Permutations

Introduction to Groups

Permutations

Subgroups

Symmetry

Equivalence and Congruence

Equivalence Relations

Congruence

Divisors and Multiples

Groups Revisited

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Generators

Rings, Integral Domains and Fields

Rings

Integral Domains and Fields

More on Rings

Isomorphism

Homomorphisms

More on Integral Domains

Ordered Ontegral Domains

Constructing Fields

The Integers

The Rational Numbers

The Real Numbers

The Complex Numbers

Others

Polynomials

Introduction

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