Abstract Algebra Ln

8/10/2019 Abstract Algebra Ln

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Chapter 1

Groups

1.1 Definitions and Examples

Definition 1.1.1. A group is a set G together with a binary operation :

G G G which satisfies the following axioms:

(i) (x y) z=x (y z) for all x, y,z inG,

(ii) e G such that e x= x = x e for allxin G,

(iii) for each xin G, y G such that y x= e = x y.

Definition 1.1.2. LetG be a group. IfG is finite, then theorderofG, denoted

|G|, is the number of elements ofG. IfG is infinite,G is said to have an infinite

order.

Definition 1.1.3. LetG be a group andx G. Thenx is said to be ofinfinite

order if there is no positive power ofx equals the identity. Ifxn = e for some

n N, the smallest such nis called the order of x, denoted o(x).

Theorem 1.1.4. LetG be a semigroup. Then TFAE:

(i) G is a group.

(ii) e G such that ae = a for all a G; and for eacha G there is an

a G such thataa =e. (e anda multiply on the right.)


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1.2 Homomorphisms 2

(iii) e G such that ea = a for all a G; and for eacha G there is an

a G such thataa= e. (e anda multiply on the left.)

Definition 1.1.5. A group G is abelian if xy = yx for all x, y G. In this

case, one can choose to write additively, that is:

(i) the binary operation is + (x + y:= x y), and

(ii) 0 is the unit element, and xdenotes the inverse ofx.

Definition 1.1.6. A nonempty subset Hof a group G is called a subgroup of

GifHis a group under the group operation inherited from G. IfHis a subgroupofG, we write HG.

Theorem 1.1.7. LetG be a group and let =HG. Then TFAE:

(i) H is a subgroup ofG.

(ii) a, b Hab H, anda Ha1 H.

(iii) a, b Hab1 H.

1.2 Homomorphisms

Definition 1.2.1. Let (G, ) and (G, ) be groups. A mapping : G G is

called a homomorphism(of groups) if

(a b) =(a) (b) for all a, b G.

A homomorphism is called anisomorphismif it is bijective (11 and onto).

Gand G are said to beisomorphic, denotedG =G, if there exists an isomor-

phism from Gonto G.

Definition 1.2.2. Let be a homomorphism from G to G. The kernel of,

ker , is defined to be

ker = {g G | (g) =e}


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1.3 Cyclic Groups and Generators 3

where e is the identity ofG.

Theorem 1.2.3. Let: G G

be a homomorphism. Then

(i) (e) =e wheree ande are identities inG andG, respectively;

(ii) (x1) = ((x))1 for allx G;

(iii) (x1x2 . . . xn) =(x1)(x2) . . . (xn) for all x1, x2, . . . , xn G;

(iv) (xn) = ((x))n for alln Z;

(v) ker is a subgroup ofG

;

(vi) ker = {e} if and only if is 1-1;

(vii) Im is a subgroup ofG.

1.3 Cyclic Groups and Generators

Theorem 1.3.1. LetG be a group anda G. Then{an | n Z} is a subgroup

ofG and it is the smallest subgroup ofG containinga.

Definition 1.3.2. The subgroup {an | n Z}of a group G is called the cyclic

subgroupofG generated by aand will be denoted by a.

Definition 1.3.3. A group G is called a cyclic group if G = a for some

a G. ais then called a generator ofG and we say that agenerates G.

Theorem 1.3.4. Every cyclic group is abelian.

Theorem 1.3.5. A subgroup of cyclic group is cyclic.

Theorem 1.3.6. LetG be a cyclic group of ordern generated bya. Letb=as

for somes < n. Thenb generates a cyclic subgroup ofG containing n

d elements,

whered= gcd(n, s).

Corollary 1.3.7. Ifa is a generator of a finite cyclic group G of ordern, then

the set of all generators ofG is{ar |gcd(n, r) = 1}.


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1.4 Group Actions 4

Theorem 1.3.8. (i) (Z, +) is the only infinite cyclic group (up to isomorphism).

(ii) (Zn, +) is the only cyclic group of order n (up to isomorphism).

Corollary 1.3.9.The subgroups of(Z, +) are the groupsnZ forn Z.

Definition 1.3.10. LetXbe a nonempty subset of a group G. The smallest sub-

group ofG containingX, denotedX, is called the subgroup of G generated

by X. We say that Xgenerates X.

IfX is finite, say X={x1, x2, . . . , xn}, we shall simply write x1, x2, . . . , xn

for X. IfX = G, we say that X is a set ofgenerators ofG. IfX is finite

and X= G, then Gis said to be finitely generated.

Theorem 1.3.11. LetG be a group andXG. Then

(i) ={e},

(ii) X= {x11 xnn | xi aren distinct elements of X, i Z andn N}.

Moreover, ifG is finite, thenX is the set of all products of elements ofX

(i.e. eachi is a positive integer).

Theorem 1.3.12. A group G is isomorphic to Dn if and only if it is generated

by two elementsa, b such thatan = 1 =b2 andbab= a1.

1.4 Group Actions

Definition 1.4.1. Let G be a group and X be a set. We say that the group

G acts on the set X or X is a Gset if there is a mapping G X X,

(g, x)g x(or gx), which satisfies

(i) 1 x= x for all x X, where 1 is the identity element ofG; and

(ii) g (h x) = (gh) x for all g, h G and all x X.

Definition 1.4.2. Let G act on X. If g = 1 is the only element of G which

fulfills the identity g x= x for all x X, then we say that G acts faithfully

on X.


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1.4 Group Actions 5

For each x X, G x = {g x | g G} is called the orbit of x and is

also denoted by orb(x). If G x = X for some x X, then G is said to act

transitively on X. For each Y X, the set {g G | g Y =Y} is called the

stabilizer ofY, denoted Stab(Y), or Stab(x) in case Y ={x}.

Theorem 1.4.3. Let G act on X. Then for each g G the function g :

X Xdefined byg(x) = g x is a permutation of X. Furthermore, the map

: G S(X) defined by(g) = g is a homomorphism with the property that

((g))(x) = g x. The homomorphism is called the permutation represen-

tationassociated to the given action. The kernel of this homomorphism is the

set{g G | g x= x x X}.

Theorem 1.4.4 (Cayleys Theorem). Every group is isomorphic to a subgroup

of a permutation group. If a group is of order n, then it is isomorphic to a

subgroup ofSn.

Theorem 1.4.5. LetG be a group. Suppose thatG acts on a setX. Define a

relation onX by

x y if and only if y= g x for someg G.

Then

(i) is an equivalence relation onX, and

(ii) theorbitofx X, G x, is its equivalence class w.r.t. the relation.

ThusXis the disjoint union of all distinct orbits under the action ofG.

Definition 1.4.6. IfH G and xG, Hx={hx | h H} is called a right

coset ofH inG, and xH={xh|hH}is called a left coset ofH in G.

Definition 1.4.7. Let H be a subgroup of a group G. Then the number of

disjoint right (or left) cosets ofH in G is called the index ofH in G, denoted

[G : H]. (i.e. [G: H] = || where is the index set of{x | }, the right

transversal ofH in G.)


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1.4 Group Actions 6

Theorem 1.4.8 (Lagranges Theorem). LetG be a finite group andHG.

Then |H| divides |G|. In particularo(g)|G| for allg G.Corollary 1.4.9. LetG be a finite group andHG. Then

|G|= [G: H] |H|.

Theorem 1.4.10. Let G be a group which acts on a setX and x X. Then

Stab(x), the stabilizer of{x}, is a subgroup ofG, and

[G: Stab(x)] =|G x|.

Theorem 1.4.11 (The Orbit-Stabilizer Theorem). LetG be a finite group

which acts on a setX. Then for eachx X,

|G|= |G x| |Stab(x)|.

Definition 1.4.12. Let G be a group and A be a nonempty subset ofG. The

centralizer of A in G, CG(A), and the normalizer of A in G, NG(A), are

defined as follows:

CG(A) = {g G | gag1 =a for all a A},

NG(A) = {g G | gAg1 =A},

In particular,Z(G) =CG(G) is called the center ofG

Theorem 1.4.13. LetG be a group and = A G. ThenCG(A) andNG(A)

are subgroups ofG. MoreoverCG(A) is a subgroup ofNG(A).

Theorem 1.4.14. Let G be a finite group and x G. Then the number ofconjugates of x is [G : CG(x)]. In particular, the number of conjugates of x

divides the group order.

Theorem 1.4.15 (The Class Equation). LetG be a finite group. Then

|G|= |Z(G)| +t

i=1

[G: CG(xi)]

wherex1, x2, . . . , xt are representatives from all orbits of size greater than one.


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1.5 Quotient Groups and Isomorphism Theorems 8

Theorem 1.5.3. Let N be a normal subgroup of a group G. Then G/N =

{N a | a G} is a group under the multiplication defined by

Na N b= Nab, (a, b G).

The group G/N is called thequotient groupor thefactor groupofG byN.

The map : G G/N defined by (a) = Na is a group homomorphism

whose kernel isN.

Definition 1.5.4. Let G H

Kbe a sequence of group homomorphisms.

We say that it is exact at H if Im = ker . A short exact sequence of

groups is a sequence of group homomorphisms

1G H

K1 ,

which is exact at G, H and K.

Theorem 1.5.5 (The First Isomorphism Theorem). If : G His a group

homomorphism, thenker is a normal subgroup ofG andG/ ker =Im .

Theorem 1.5.6. IfH andKare finite subgroups of a group, |HK|= |H||K|

|H K|.

Theorem 1.5.7. IfHandKare subgroups of a groupG, thenH Kis a subgroupif and only ifHK=K H.

Corollary 1.5.8. If H and K are subgroups of a group G and H NG(K),

thenHK is a subgroup ofG. In particular, ifK is normal inG, thenHK is a

subgroup ofG for any subgroup H ofG.

Theorem 1.5.9 (The Second Isomorphism Theorem). Let H and N be

subgroups of a group G withNnormal. ThenHNis normal inHandH

(H

N)=HN/N.

Theorem 1.5.10 (The Third Isomorphism Theorem). LetN be a normal

subgroup of a group G. Then the map H H/N gives a 1-1 correspondence

between the set of subgroups ofG containingNand the set of subgroups ofG/N.

Moreover this correspondence carries normal subgroups to normal subgroups. If

H G, andNHG, then

G/H=(G/N)(H/N).


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1.6 Direct Products and Abelian Groups 9

1.6 Direct Products and Abelian Groups

Definition 1.6.1. Let {A | } be a family of groups. The

set

A is a group under the coordinatewise operation. It is

called the (strong) direct product of the groups A. The subgroup(a)

A | all but finitely many a= 1

is called the weak direct

product (or direct sum) of the groups A, and is denoted

{A | }or

A.

Theorem 1.6.2. IfA andB are subgroups ofG such that

(i) A B={e},

(ii) AB=G, and

(iii) ab= ba aA bB ,

thenG = A B. We say thatG is the(internal) direct productofA and

B. Note that the condition (iii) can be replaced by that A and B are normal

subgroups ofG.

Theorem 1.6.3 (Chinese Remainder Theorem). Letm1, . . . , mk, n1, . . . , nk

be integers. Ifm1, m2, . . . , mk are pairwise relatively prime, then there exists an

integern such that

n ni mod mi for all i= 1, . . . , k.

Theorem 1.6.4. Let m1, m2, . . . , mk be pairwise relatively prime integers and

m = m1m2 mk. If g is an element in G satisfying gm = 1, then there exist

uniqueg1, g2, . . . , gkG such that

(i) gmii = 1 for alli= 1, 2, . . . , k,

(ii) g1, g2, . . . , gk are pairwise commutative, and

(iii) g= g1g2 . . . gk.


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1.6 Direct Products and Abelian Groups 10

Definition 1.6.5. Let gbe an element of a group G. Ifghas order m= p11 pkk

where pis are distinct primes, then pi is called the piprimary part ofg.

Theorem 1.6.6. Let A be a finite abelian group of order m = m1m2 . . . mk

where themis are pairwise relatively prime. LetAi ={g A | gmi = 1}. Then

A=A1 A2 Ak. Moreover|Ai|= mi.

Theorem 1.6.7. LetA be an abelian group of orderp, wherep is a prime. If

A has exponentp (i.e. ap = 1 for alla A), then

A= Zp Zp Zp times

= (Zp).

Definition 1.6.8. A group G is called a p-group, where p is a prime, if every

element ofG has the order as a (finite) power ofp.

Definition 1.6.9. A positive integer n is said to be an exponentof a group G,

ifgn = 1 for each g G. In this caseG is said to have finite exponent, and

the least such positive integer n is called the exponent ofG.

Theorem 1.6.10 (Burnside Basis Theorem for Abelian p-Group). LetA

be an abelian group of exponentp wherep is a prime. IfH is a subgroup ofA

andHAp =A, thenH=A. (Equivalence: If the cosetsApa1, . . . , Apak ofA/A

p

generateA/Ap, thena1, a2, . . . , ak generateA.)

Theorem 1.6.11. If A is a finite abelian group of exponent p, where p is a

prime, then for anyHA, there exists a subgroup KofA such thatA =HK.

(Equivalence: IfVis a finite dimensional vector space overZpandUis a subspace

ofV, then there is a subspaceW ofV such thatV =U W.)

Theorem 1.6.12. Every finite abelian pgroup is a direct product of cyclic

groups.

Theorem 1.6.13. LetA= (Zp)u1 (Zp2)

u2 . . . (Zpm)um, and

B= (Zp)v1 (Zp2)

v2 . . . (Zpm)vm whereui 0, vj 0.

IfA=B, thenui=vi for alli= 1, 2, . . . , m .


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1.7 The Sylow Theorem and Applications 12

Theorem 1.7.4 (The Sylow Theorem). LetG be a group of orderpmwhere

p is a prime, >0, andp m. Then

(i) G contains a Sylowp-subgroup.

(ii) The number of Sylowp-subgroups is 1 modp.

(iii) IfH is ap-subgroup ofG andP is a Sylowpsubgroup ofG, then some

conjugate ofP containsH. In particular,

(iv) All Sylowp-subgroups ofG are conjugated.

Corollary 1.7.5. The number of Sylowp-subgroup ofG divides |G|. (In partic-

ular it dividesm if|G|= pm, p m).

Corollary 1.7.6. LetG be a finite group andSa Sylowp-subgroup ofG. Then

Sis the only Sylowp-subgroup ofG if and only ifS G.

Theorem 1.7.7. Letn= pm wherep is a prime andpm. Then

n

p

m

mod p.

Theorem 1.7.8 (Combinatorial Proof of The Sylows Theorem ). Let

|G|= pam, wherep is a prime andp m. ThenG has a subgroup of orderpa.

Theorem 1.7.9. LetG be a finite group andP a Sylowp-subgroup ofG. Then

(i) N(P) is equal to its own normalizer, i.e. N(N(P)) =N(P).

(ii) IfN(P) T G, thenT is equal to its own normalizer, i.e. N(T) =T .

Theorem 1.7.10. LetG be a group and|G|= p

mwherep is a prime andp m.ThenGhas a subgroup of orderp for eachwhere1 . Moreover, every

subgroup HofG of orderp is normal in a subgroup of orderp+1 for1 .

Definition 1.7.11. An automorphism of a group G is an isomorphism : G

G.The set of all automorphism of a group Gis denoted Aut(G).

Theorem 1.7.12. (i) With group operation the composition of functions,

Aut(G) is a group.


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1.7 The Sylow Theorem and Applications 13

(ii) Each g G determines an automorphism on G, g : G G defined by

(g)(h) = ghg1. g is called an inner automorphism. The subgroup

of Aut(G) consisting of all such g, {g | g G} is called the inner

automorphism groupofG and is denotedInn(G).

(iii) The map gg is a group homomorphismG Aut(G).

(iv) The kernel of isZ(G), i.e. the center ofG. The image of is Inn(G).

Definition 1.7.13. A subgroup Hof a group G is a characteristic subgroup

if(H) =H for each Aut(G).

Definition 1.7.14. A group issimple if it has no nontrivial normal subgroups.


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2.2 Ring Homomorphisms and Quotient Rings. 15

Definition 2.1.6. A ring D with identity is a division ring (or skew field)

if every nonzero element inD is invertible.

Definition 2.1.7. A commutative division ring is called a field.

Theorem 2.1.8. Every field is an integral domain.

Theorem 2.1.9. Every finite integral domain is a field.

Definition 2.1.10. A subset SR is a subringof a ring R ifSis a ring with

respect to + and in R.

Theorem 2.1.11. LetSbe a nonempty subset of a ringR. ThenS is a subring

ofR if and only ifa b andab S for alla, b S.

2.2 Ring Homomorphisms and Quotient Rings.

Definition 2.2.1. Let R and Sbe rings. :R S is called a (ring) homo-

morphism if

(x + y) =(x) + (y) and (xy) =(x)(y) x, y R.

Thekernel of, denoted ker is the set {x R | (x) = 0}. Anisomorphism

is a bijective homomorphism.

Definition 2.2.2. LetAbe a subring of a ringR. ThenAis called aleft(right)

ideal ofR ifRA A (AR A). Ais an idealofR ifA is both a left and right

ideal ofR.

Theorem 2.2.3. LetRbe a ring andIan ideal ofR. ThenR/I is a ring underthe operations

(r+ I) + (s + I) = (r+ s) + I

(r+ I)(s + I) =rs+ I.

It is called the quotient (or factor) ring ofR byI.

The map : R R/I defined by (r) = r+I is a ring homomorphism

which is surjective and has the kernelI. is called thecanonical (or natural)

projection ofR onto RI.


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2.2 Ring Homomorphisms and Quotient Rings. 16

Theorem 2.2.4. Let: R S be a ring homomorphism. Then

(i) Im is a subring ofS,

(ii) ker is an ideal ofR, and

(iii) R/ ker =Im .

Theorem 2.2.5 (The Second Isomorphism Theorem). LetSbe a subring

ofR and letIbe an ideal ofR. Then

(i) S+ I is a subring ofR,

(ii) S I is an ideal ofS, and

(iii) (S+ I)/I=S/(S I).

Theorem 2.2.6 (The Third Isomorphism Theorem). Let Ibe an ideal of

R and A be an ideal of R containing I. Then A

I is an ideal of R

I and

(R

I)

(A

I)=R

A.

Theorem 2.2.7. Let I be an ideal ofR. The correspondence A A/I is an

inclusion preserving bijection between the set of subringsA ofR containingIand

the set of subrings ofR/I. Furthermore, a subringA containingI is an ideal of

R if and only ifA/I is an ideal ofR/I.

Theorem 2.2.8. A ringR with 1 is a division ring if and only if 0 andR are

the only left (right ) ideals ofR.

Corollary 2.2.9. LetR be a commutative ring with 1. ThenR is a field if andonly if its only ideals are 0 andR.

Corollary 2.2.10. IfR is a field then any nonzero ring homomorphism fromR

into another ring is an injection.


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2.3 Properties of Ideals. 17

2.3 Properties of Ideals.

From now on R is a ring with identity 1= 0.

Definition 2.3.1. LetAbe a nonempty subset of a ringR.Theideal generated

byA, denoted (A) is the smallest ideal ofR containingA. An ideal generated by

a single element set,{a}, is called a principal ideal, it will be denoted (a).

IfA= {a1, a2, . . . , an},the ideal generate byAis called afinitely generated

idealand denoted (a1, a2, . . . , an).

RA= {r1a1+ r2a2+ . . . + rn

an

| ri R, a

i A, n N}

is the left ideal generated by A.

AR= {a1r1+ a2r2+ . . . + anrn | ri R, ai A, n N}

is the right ideal generated by A.

RAR= {r1a1s1+ r2a2s2+ . . . + rnansn | ri, si R, ai A, n N}

is the ideal generated by A.

Definition 2.3.2. An idealMof a ringR is called a maximal ideal ifM=R

and the only ideals containing M are M and R.

Theorem 2.3.3. Every proper ideal in a ring with identity1= 0 is contained in

a maximal ideal.

Theorem 2.3.4. LetRbe a commutative ring andMan ideal ofR. ThenM is

an maximal ideal if and only ifR/M is a field.

Definition 2.3.5. Let R be a ring and Pan ideal ofRwith P =R. P is called

aprime ideal if wheneverA and B are ideals ofR and AB P, thenA P or

B P. If, in addition,R is commutative, then this notion becomes the notion

of prime in Z.

Lemma 2.3.6. An idealPof a ringR is prime if and only if for anyaandbin

R, aRb P impliesa P orb P.


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2.3 Properties of Ideals. 18

Theorem 2.3.7. LetRbe a commutative ring andPan ideal ofRwithP=R.

ThenP is prime if and only ifab P impliesa P orb P for anya, b R.

Theorem 2.3.8. LetPbe an ideal of a commutative ringR. ThenPis a prime

ideal if and only ifR/Pis an integral domain.

Corollary 2.3.9.Every maximal ideal of a commutative ring is a prime ideal.

Definition 2.3.10. Let R be a commutative ring with 1 and x, y R. We

say that x divides y, denoted x|y if there exists qR such that y = xq (i.e.

Ry Rx or (y)(x)).

Definition 2.3.11. Let R be an integral domain and a, b R. We say that a

and b are associated ifa| b and b| a.

Theorem 2.3.12. LetR be an integral domain, a, b R. Then TFAE:

(i) a andb are associated,

(ii) Ra= Rb, and

(iii) a= ub for someu U(R).

Definition 2.3.13. Let R be an integral domain and a, b R. A greatest

common divisor ofaand bis an element d which satisfies:

(i) d| a and d| b, and

(ii) ifd1| a and d1| b, then d1| d.

A least common multiple ofaand b is an element m which satisfies:

(i) a| m and b| m, and

(ii) ifa| m1 and b| m1, then m| m1.

Theorem 2.3.14. LetR be an integral domain anda, b R.

(i) Ifd andd1 are gcds ofa andb, thend andd1 are associated.


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2.4 Euclidean Domains 19

(ii) Ifm andm1 are lcms ofa andb, thenm andm1 are associated.

Theorem 2.3.15. Let R be an integral domain. If Ra+ Rb = Rc, then c =gcd(a, b).

Definition 2.3.16. Let Rbe an integral domain. A nonzero element pis called

a prime inR ifRp is a prime ideal.

Definition 2.3.17. Let R be an integral domain. An elementa in R is called

an irreducible element(atom) if

(i) a= 0 and a /U(R), and

(ii) a cannot be expressed as a product a= bc where b /U(R), c /U(R).

Theorem 2.3.18. Every prime element in an integral domain is irreducible.

2.4 Euclidean Domains

Definition 2.4.1. A function N : R N {0} = N0 is called a normon an

integral domain R ifN(0) = 0.

A normN is said to be multiplicative if it satisfies the following conditions:

(i) N(a) = 0 if and only ifa= 0.

(ii) N(ab) =N(a)N(b) for alla, b R.

Proposition 2.4.2. LetR be an integral domain with a multiplicative normN

onR. Then

(i) N(u) = 1 for every unitu inR.

(ii) If in additionNhas a property that everyx such thatN(x) = 1 is a unit

inR, then an element inR, withN() =p for some primepinZ, is an

irreducible element ofR.

Definition 2.4.3. Let R be an integral domain. R is said to be a Euclidean

Domain if there is a function N from R {0}to Nsatisfying


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2.5 Principal Ideal Domains 20

(i) N(ab) N(a) for all nonzero elementsa and bin R.

(ii) Ifa, b R and b = 0, then there exist q, r R such that a= bq+ r withr= 0 or r = 0 andN(r)< N(b).

Proposition 2.4.4. LetR be a Euclidean Domain with normN. Then

(i) N(1) is minimal amongN(a) for all nonzero a R.

(ii) U(R) ={u R | N(u) = 1}.

Theorem 2.4.5. Z[i] is a Euclidean Domain.(with respect to the normN(a+

bi) =a2 + b2.)

Theorem 2.4.6. Every ideal of a Euclidean domain is a principal ideal.

Theorem 2.4.7 (Euclidean Algorithm). Let R be a Euclidean domain and

leta andb be elements inR. Then

Ra + Rb= Rc

for some c R. Furthermore, c can be explicitly constructed and gcd(a, b) = c

so gcd(a, b) always exists.

2.5 Principal Ideal Domains

Definition 2.5.1. A Principal Ideal Domain (PID) is an integral domain

in which every ideal is principal.

Theorem 2.5.2. Every nonzero prime ideal in a PID is a maximal ideal.

Theorem 2.5.3. In a PID, (p) is a maximal ideal if and only ifp is irreducible.

Corollary 2.5.4. LetR be a PID andp R. Thenp is irreducible if and only

if(p) is a prime ideal.

Theorem 2.5.5. In a PID, a nonzero element is a prime if and only if it is

irreducible.


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2.7 Fields of Fractions 22

(ii) Every nonzero nonunit element ofR can be expressed uniquely (up to or-

dering) as a = up11 pkk , where u is a unit k 0, 1, , k > 0 and

p1, . . . , pk are distinct elements ofP.

Theorem 2.6.5. LetR be a UFD anda, b R. Then

(i) a andb have a gcd.

(ii) a andb have an lcm.

(iii) IfP is a set of representative irreducible elements forR, then among the

gcd of a and b, there is exactly one which is a product of elements of P.

The same is true for lcm.

(iv) Ifa and b are nonzero, gcd(a, b) = d and lcm(a, b) = m thenab anddm

are associated.

Theorem 2.6.6. Every PID is a UFD. In particular, every Euclidean domain is

a UFD.

2.7 Fields of Fractions

Theorem 2.7.1. Let R be an integral domain. Define a relation on S =

R (R \ {0}) by

(r1, s1) (r2, s2)r1s2 = r2s1.

Then

(i) is an equivalence relation onS.

(ii) Q(R) =S/ is a field under the following addition and multiplication:

[(r1, s1)] + [(r2, s2)] = [(r1s2+ r2s1, s1s2)],

[(r1, s1)] [(r2, s2)] = [(r1r2, s1s2)].

(iii) Q(R)is the smallest field containingR in the sense that any field containing

an isomorphic copy ofR in which all nonzero elements ofR are units must

contain an isomorphic copy ofQ(R).


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2.8 Polynomial Rings 23

Definition 2.7.2. The field in Theorem 2.7.1 is called the field of fractions

or quotient field ofR.

2.8 Polynomial Rings

Definition 2.8.1. The polynomial ring R[x] in the indeterminate x with co-

efficient fromRis the set of formal sums of the form

f(x) =anxn + an1x

n1 + + a1x + a0,

where n 0, a0, . . . , an R with an = 0. The integer n is called the degree of

f. The degree of 0 is defined to be . The polynomialfis called monic if

an = 1.

Define the addition and the multiplication on R[x] as follows:

(

ni=0

aixi) + (

ni=0

bixi) =

ni=0

(ai+ bi)xi

(n

i=0

aixi) (

n

i=0

bixi) =

m+n

k=0

(k

i=0

aibki)xk,

where some leading terms ai or bj are allowed to be zero.

ThenR[x] is a ring with identity 1. IfR is commutative then so is R[x]. Note

that Rcan be considered as a subring ofR[x].

Theorem 2.8.2. LetR be an integral domain(entire ring)andf(x), g(x) R[x].

Then

(i) deg(f(x) +g(x)) max{deg f(x), deg g(x)} (note that the hypothesis thatR is an integral domain is unnecessary).

(ii) deg(f(x)g(x)) = degf(x) + degg(x).

(iii) U(R[x]) =U(R).

(iv) R[x] is an integral domain. (entire ring).


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Theorem 2.8.3. Let R be a commutative ring with 1 and i : R R[x] the

inclusion map. Let S be a commutative ring with 1 and : R S a ring

homomorphism. Then there exists a unique homomorphism : R[x] S such

that(x) = a where a S and = i. In particular, if R = S, then

: R[x] R is given by(f(x)) = f(a) andker = R[x](x a), the ideal of

R[x] generated byx a.

Definition 2.8.4. The polynomial ring in the variables x1, x2, . . . , xn with

coefficients in R is denoted by R[x1, . . . , xn] and defined inductively by

R[x1, x2, . . . , xn1][xn].

Theorem 2.8.5 (Division Algorithm). LetR be a ring with 1 (not necessary

commutative). Let f(x) be amonic polynomial of degree n in R[x]. Then for

anyg(x) R[x], there exist unique polynomialsq(x) andr(x) inR[x] satisfying

(i) g(x) =f(x)q(x) + r(x).

(ii) degr(x)< n.

Theorem 2.8.6. LetR be a commutative ring with 1, aR andf(x)R[x].

Then

(i) g(x) R[x], f(x) = (x a)g(x) + f(a).

(ii) (x a)|f(x)f(a) = 0.

Definition 2.8.7. Let f(x) R[x] anda R. Then ais called a root off(x) if

f(a) = 0.

Theorem 2.8.8. Let R be an integral domain and f(x) R[x]\{0}. If

a1, a2, . . . , ak are distinct roots off(x), then(x a1)(x a2) (x ak)|f(x).

Theorem 2.8.9. IfF is field, thenF[x] is a Euclidean Domain.

Theorem 2.8.10. LetF be a field.

(i) Iff(x) F[x] anddeg f=n, thenf(x) has at mostn distinct roots.


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(ii) Iff(x), g(x) F[x] and deg f, deg g n, andf(i) = g(i) for all i =

1, 2, . . . , n+ 1 whereis are distinct elements inF, thenf(x) =g(x).

Definition 2.8.11. Let f(x) R and a a root off(x). Iff(x) is divisible by

(x a)m but not by (x a)m+1 for some positive integer m, then a is said to be

a root of multiplicity m.

Corollary 2.8.12. IfF is a field, f(x) F[x], anddegf(x) =n, thenf(x) has

at mostn roots.

Theorem 2.8.13. LetFbe a field withqelements. Then

(i) F =F\{0} is a cyclic group (under multiplication) of orderq 1.

(ii) IfF ={a1, . . . , aq1}, then(x a1)(x a2) (x aq1) =xq1 1.

(iii) IfF={0, a1, . . . , aq1}, thenx(x a1)(x a2) (x aq1) =xq x.

Theorem 2.8.14. LetF be a field. Then

(i) Linear polynomials(polynomial of degree 1) are irreducible inF[x].

(ii) Linear polynomials are the only irreducible elements inF[x] iff each poly-

nomial of positive degree has a root inF.

Definition 2.8.15. LetR be a UFD and f(x) =anxn + an1x

n1 + + a1+ a0

is a nonzero polynomial over R. The content off(x) is the greatest common

divisor ofa0, a1, . . . , an. We say that f(x) is primitive ifa0, a1, . . . , an have no

common divisor except units.

Theorem 2.8.16. LetR be a UFD andf(x), g(x) R[x]. Iff(x) andg(x) are

primitive, then so isf(x)g(x).

Theorem 2.8.17. LetR be a UFD andf(x) andg(x) are nonzero polynomials

ofF[x]. Then

(i) f(x) is primitive iff the content off(x) is 1.

(ii) Ifa is the content off, thenf(x) =af1(x) wheref1(x) is primitive.


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(iii) Iff(x) =af1(x) wheref1(x) is primitive, thena is a content off.

(iv) Ifa is the content off(x)andb is the content ofg(x), thenab is the contentoff g.

Theorem 2.8.18. Let R be a UFD and F = Q(R) be its field of fraction. If

f(x) is an irreducible polynomial inR[x], thenf(x), considered as a polynomial

in F[x] is irreducible inF[x]. In particular, if f(x) Z[x] is irreducible inZ,

thenf(x) is irreducible overQ.

Theorem 2.8.19. Let R be a UFD and F its field of quotient. Let f(x) =

anxn + an1x

n1 + + a1x + a0 R[x]. If r

sF is a root off(x), wherer and

s are relatively prime, then(sx r)|f(x) inR[x] and so s|an andr|a0 ifr= 0.

Corollary Let R be a UFD and F its field of fractions. Let f(x) R[x] be

primitive. Then f(x) is irreducible in R[x] ifff(x) is irreducible in F[x].

Theorem 2.8.20 (Eisensteins Criterion). LetPbe a prime ideal of the in-

tegral domainR and letf(x) =xn +an1xn1 + +a1x+a0 be a polynomial

inR[x] (n1). Assume thatan1, an2, . . . , a1, a0 are all inP anda0 is not in

P2.Thenf(x) is irreducible inR[x].

Corollary (Eisensteins Criterion for Z[x]) Let p be a prime in Z and let

f(x) =xn + an1xn1 + + a1x + a0 Z[x](n 1). Ifp|ai for all i but p2 a0,

then f(x) is irreducible in both Z[x] and Q[x].

Theorem 2.8.21. LetR be UFD andF its field of fractions. Letf(x)R[x].

Thenf(x) is irreducible inR[x] iff either

(i) f(x) R andf(x) is irreducible inR, or

(ii) f(x) is a primitive polynomial of degree n 1 andf(x) is irreducible in

F[x].

Theorem 2.8.22. Let R be a UFD and f(x) R[x]. If f(x) is irreducible in

R[x], then(f(x)) is a prime ideal, i.e. f(x) is a prime.


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Theorem 2.8.23. IfR is a UFD, thenR[x] is a UFD.

Corollary If R is a UFD then a polynomial ring in an arbitrary number ofvariables with coefficient in Ris also a UFD.


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Chapter 3

Fields

3.1 The Characteristic Fields

Definition 3.1.1. The characteristic of a field F, denoted char(F), is the

smallest positive integer m with the property that m 1 = 0 provided such a m

exists, otherwise, it is defined to be 0.

Proposition 3.1.2. For any fieldF, char(F) is either0 or a primep.

3.2 Extension Fields and Degrees of Extensions

Definition 3.2.1. A fieldKis said to be an extension fieldof a fieldFand is

denoted K|F, ifK F. The dimension ofKas a vector space over F is called

the degree of a field extension K|F, denoted [K : F]. The extension is said

to be finite if [K :F] isfinite and it is said to be infinite otherwise.

Theorem 3.2.2. If K|E and E|F are finite field extensions, then K|F is also a

finite field extension and

[K :F] = [K :E][E:F]

.


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3.2 Extension Fields and Degrees of Extensions 29

Theorem 3.2.3. LetFbe a field andp(x) F[x] be an irreducible polynomial.

Then there is an extension fieldE ofF in whichp(x) has a root and [E :F] =

degp(x). Moreover

E={bn1n1+bn2

n2+ +b1+b0| bn1, bn2,...,b1, b0 F} where = x+(p(x)).

Definition 3.2.4. Let K|Fbe a field extension of a field F and 1, 2, . . . , n

K. The intersection of all subfields of K containing F and 1, 2, . . . , . . . , n,

is called the field generated by 1, 2, . . . , . . . , n over F and is denoted

F(1, 2, . . . , n).

It is the smallest subfield with the above property. In particular, for each

K, F() is the smallest subfields ofKcontaining F and .

Lemma 3.2.5. LetK|Fbe a field extension of a fieldF. Let, K. Then

F(, ) = (F())().

Moreover, if[F() :F] =m and[F()() :F()] =n, then[F(, )] =mn and

any element ofF(, ) has the form i=1,2,...,mj=1,2,...,n

aijij.

Definition 3.2.6. Let E1 and E2 be subfields ofE. Then the composite field

ofE1andE2, denotedE1E2, is the smallest subfield ofEcontaining bothE1 and

E2. In general, the composite of any collection of subfields ofE is the smallest

subfield ofEcontaining all the subfields.

Theorem 3.2.7. LetE1 and E2 be two finite field extensions of a filedF both

contained inE. Then [E1E2 : F] [E1:F][E2 : F].

CorollaryLetE1 andE2 be two finite field extensions of a fieldFboth contained

in E. Assume [E1 : F] = m and [E2 : F] = n where (m, n) = 1. Then

[E1E2:F] = [E1 : F][E2:F] =mn

Theorem 3.2.8. Let p(x) be an irreducible polynomial of F[x]. Let K be an

extension field ofF containing a root ofp(x). Then

(i) F()=F[x]/(p(x)).


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3.3 Algebraic Extensions 30

(ii) Ifdegp(x) =n, thenF() ={a0+ a1+ + an1n1 |ai F}.

Theorem 3.2.9. Let: FF

be a field isomorphism. Then induces a ringisomorphismF[x] F[x] with the property that(a) = (a) for alla F,

and maps a irreducible polynomial to the a irreducible polynomial. Moreover,

if is a root of an irreducible polynomialp(x) and is a root of(p(x)), then

there exists an isomorphism:F() F

() mapping to and extending.

3.3 Algebraic Extensions

Definition 3.3.1. Let E|Fbe a field extension. E is said to be algebraic

over F if is a root of some polynomial f(x) F[x]. If is not algebraic over

F, then is said to be transcendental over F. E|F is said to bealgebraic if

every element ofE is algebraic over F, andEis called an algebraic extension

ofF.

Theorem 3.3.2. Let E|F be an extension and assume that E is algebraic

overF. Then

(i) ! monic irreducible polynomial, denotedmF(), which has as a root.

(ii) f(x) F[x] has as a root iff mF() divides f(x) in F[x] (i.e. if I =

{f(x) F[x]| f() = 0}, thenIis the ideal generated bymF()).

(iii) F()=F[x]/(mF()) and [F() :F] = degmF().

(iv) F[] =F().

Definition 3.3.3. The unique monic irreducible polynomial mF() in theorem

3.3.2 is called the minimal polynomial for overF. The degree ofmF() is

called the degree ofover F, denoted deg(, F)

Theorem 3.3.4. LetE|F be an extension and E. Then is algebraic over

F iffF[] is the fieldF(), whereF[] ={f()| f(x) F[x]}.

Theorem 3.3.5. LetE|F be an extension. If [E : F]


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3.4 Finite Fields 31

Theorem 3.3.6. LetK|F be an extension. Then [K :F] is finite if and only if

K=F(1, 2,...,k) for some algebraic elements1, 2,...,k overF. Moreover

if, for eachi, [F(i) :F] =ni, then [K :F] n1n2 nk.

Corollary Let K|F be an extension. Then the set of elements of K that are

algebraic overF forms a subfield ofK.

Theorem 3.3.7. IfK|E andE|Fare algebraic extensions, then so isK|F.

3.4 Finite Fields

Theorem 3.4.1. Every finite field must have prime power order.

Corollary 3.4.2. Every element of a finite field with characteristicp is algebraic

overZp.

Definition 3.4.3. The extension field KofF is called a splitting field for the

polynomial f(x) F[x] if f(x) factors completely into linear factors (or splits

completely) inK[x] and f(x) does not factor completely into linear factors overany proper subfield ofKcontaining F.

Theorem 3.4.4. For any fieldF, iff(x) F[x], then there exists an extension

K ofFwhich is a splitting field forf(x).

Theorem 3.4.5. Let:F F be an isomorphism of fields. Letf(x) F[x]

and letf(x) F[x] be the polynomial obtained by applying to the coefficients

off(x).LetEbe a splitting field forf(x)overFand LetE be a splitting field for

f(x) overF. Then the isomorphism extends to an isomorphism: EE.

Theorem 3.4.6. (Uniqueness of Splitting Fields) Any two splitting fields for a

polynomialf(x) F[x] over a fieldF are isomorphic.

Theorem 3.4.7. For each primep and each positive integern, there is (up to

isomorphism) a unique finite field of orderpn.


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3.5 Simple Extensions 32

3.5 Simple Extensions

Definition 3.5.1. LetK|Fbe a field extension. Kis called asimple extension

ofF ifK=F() for some Kand this is called a primitive element for the

extension.

Theorem 3.5.2 (Artin). LetE|F be a finite degree field extension. ThenE=

F() for some E if and only if there are only finitely many field K with

F KE.

Corollary 3.5.3. LetK|Fbe a field extension. Assume thatK=F() for some which is algebraic overF. ThenEis a simple extension for any fieldE such

thatFEK.

Theorem 3.5.4. IfF is a field of characteristic0 and if andare algebraic

overF, then there is F(, ) such thatF(, ) =F().

Abstract Algebra Ln

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