This section provides an overview of basic algebraic objects and their properties, for refer-ence in the remainder of this handbook. Several of the definitions in§2.5.1 and§2.5.2 were presented earlier in§2.4.3 in the more concrete setting of the algebraic structureZ∗n. 2.161 Definition Abinary operation∗on a setSis a mapping fromS×StoS. That is,∗is a
rule which assigns to each ordered pair of elements fromSan element ofS.
2.5.1 Groups
2.162 Definition Agroup(G,∗)consists of a setGwith a binary operation∗onGsatisfying the following three axioms.
(i) The group operation isassociative. That is,a∗(b∗c) = (a∗b)∗cfor alla, b, c∈G.
(ii) There is an element1∈G, called theidentity element, such thata∗1 = 1∗a=a for alla∈G.
(iii) For eacha∈Gthere exists an elementa−1∈G, called theinverseofa, such that a∗a−1=a−1∗a= 1.
A groupGisabelian(orcommutative) if, furthermore, (iv) a∗b=b∗afor alla, b∈G.
Note that multiplicative group notation has been used for the group operation. If the group operation is addition, then the group is said to be anadditivegroup, the identity ele-ment is denoted by0, and the inverse ofais denoted−a.
Henceforth, unless otherwise stated, the symbol∗will be omitted and the group oper-ation will simply be denoted by juxtaposition.
2.163 Definition A groupGisfiniteif|G|is finite. The number of elements in a finite group is called itsorder.
2.164 Example The set of integersZwith the operation of addition forms a group. The identity element is0and the inverse of an integerais the integer−a.
2.165 Example The setZn, with the operation of addition modulon, forms a group of order n. The setZnwith the operation of multiplication modulonis not a group, since not all elements have multiplicative inverses. However, the setZ∗n(see Definition 2.124) is a group of orderφ(n)under the operation of multiplication modulon, with identity element1.
2.166 Definition A non-empty subsetHof a groupGis asubgroupofGifHis itself a group with respect to the operation ofG. IfHis a subgroup ofGandH=G, thenHis called a propersubgroup ofG.
2.167 Definition A groupGiscyclicif there is an elementα∈Gsuch that for eachb∈Gthere is an integeriwithb=αi. Such an elementαis called ageneratorofG.
2.168 Fact IfGis a group anda∈G, then the set of all powers ofaforms a cyclic subgroup of G, called the subgroupgenerated bya, and denoted bya.
2.169 Definition LetGbe a group anda∈G. Theorderofais defined to be the least positive integertsuch thatat = 1, provided that such an integer exists. If such atdoes not exist, then the order ofais defined to be∞.
2.170 Fact LetGbe a group, and leta∈Gbe an element of finite ordert. Then|a|, the size of the subgroup generated bya, is equal tot.
2.171 Fact (Lagrange’s theorem) IfGis a finite group andHis a subgroup ofG, then|H|divides
|G|. Hence, ifa∈G, the order ofadivides|G|.
2.172 Fact Every subgroup of a cyclic groupGis also cyclic. In fact, ifGis a cyclic group of ordern, then for each positive divisordofn,Gcontains exactly one subgroup of orderd.
2.173 Fact LetGbe a group.
(i) If the order ofa∈Gist, then the order ofakist/gcd(t, k).
(ii) IfGis a cyclic group of ordernandd|n, thenGhas exactlyφ(d)elements of order d. In particular,Ghasφ(n)generators.
2.174 Example Consider the multiplicative groupZ∗19={1,2, . . . ,18}of order18. The group is cyclic (Fact 2.132(i)), and a generator isα= 2. The subgroups ofZ∗19, and their
gener-ators, are listed in Table 2.7.
Subgroup Generators Order
{1} 1 1
{1,18} 18 2
{1,7,11} 7,11 3
{1,7,8,11,12,18} 8,12 6 {1,4,5,6,7,9,11,16,17} 4,5,6,9,16,17 9 {1,2,3, . . . ,18} 2,3,10,13,14,15 18
Table 2.7:The subgroups ofZ∗19.
2.5.2 Rings
2.175 Definition Aring(R,+,×)consists of a setRwith two binary operations arbitrarily de-noted+(addition) and×(multiplication) onR, satisfying the following axioms.
(i) (R,+)is an abelian group with identity denoted0.
(ii) The operation×is associative. That is,a×(b×c) = (a×b)×cfor alla, b, c∈R.
(iii) There is a multiplicative identity denoted1, with1= 0, such that1×a=a×1 =a for alla∈R.
(iv) The operation×isdistributiveover+. That is,a×(b+c) = (a×b) + (a×c)and (b+c)×a= (b×a) + (c×a)for alla, b, c∈R.
The ring is acommutative ringifa×b=b×afor alla, b∈R.
2.176 Example The set of integersZwith the usual operations of addition and multiplication is
a commutative ring.
2.177 Example The setZnwith addition and multiplication performed modulonis a
commu-tative ring.
2.178 Definition An elementaof a ringRis called aunitor aninvertible elementif there is an elementb∈Rsuch thata×b= 1.
2.179 Fact The set of units in a ringRforms a group under multiplication, called thegroup of unitsofR.
2.180 Example The group of units of the ringZnisZ∗n(see Definition 2.124).
2.5.3 Fields
2.181 Definition Afieldis a commutative ring in which all non-zero elements have multiplica-tive inverses.
2.182 Definition Thecharacteristicof a field is0if
mtimes
1 + 1 +· · ·+ 1is never equal to0for any m ≥1. Otherwise, the characteristic of the field is the least positive integermsuch that m
i=11equals0.
2.183 Example The set of integers under the usual operations of addition and multiplication is not a field, since the only non-zero integers with multiplicative inverses are1and−1. How-ever, the rational numbersQ, the real numbersR, and the complex numbersCform fields
of characteristic0under the usual operations.
2.184 Fact Znis a field (under the usual operations of addition and multiplication modulon) if and only ifnis a prime number. Ifnis prime, thenZnhas characteristicn.
2.185 Fact If the characteristicmof a field is not0, thenmis a prime number.
2.186 Definition A subsetF of a fieldEis asubfieldofEifF is itself a field with respect to the operations ofE. If this is the case,Eis said to be anextension fieldofF.
2.5.4 Polynomial rings
2.187 Definition IfRis a commutative ring, then apolynomialin the indeterminatexover the ringRis an expression of the form
f(x) =anxn+· · ·+a2x2+a1x+a0
where eachai ∈ Randn ≥ 0. The elementaiis called thecoefficientofxiinf(x).
The largest integermfor whicham = 0is called thedegreeoff(x), denoteddegf(x);
amis called theleading coefficientoff(x). Iff(x) = a0(aconstant polynomial) and a0= 0, thenf(x)has degree0. If all the coefficients off(x)are0, thenf(x)is called the zero polynomialand its degree, for mathematical convenience, is defined to be−∞. The polynomialf(x)is said to bemonicif its leading coefficient is equal to1.
2.188 Definition IfRis a commutative ring, thepolynomial ringR[x]is the ring formed by the set of all polynomials in the indeterminatexhaving coefficients fromR. The two opera-tions are the standard polynomial addition and multiplication, with coefficient arithmetic performed in the ringR.
2.189 Example (polynomial ring) Letf(x) =x3+x+ 1andg(x) =x2+xbe elements of the polynomial ringZ2[x]. Working inZ2[x],
f(x) +g(x) =x3+x2+ 1 and
f(x)·g(x) =x5+x4+x3+x.
For the remainder of this section,Fwill denote an arbitrary field. The polynomial ring F[x]has many properties in common with the integers (more precisely,F[x]andZare both Euclidean domains, however, this generalization will not be pursued here). These similar-ities are investigated further.
2.190 Definition Letf(x)∈F[x]be a polynomial of degree at least1. Thenf(x)is said to be irreducible overF if it cannot be written as the product of two polynomials inF[x], each of positive degree.
2.191 Definition (division algorithm for polynomials) Ifg(x), h(x) ∈ F[x], withh(x) = 0, then ordinary polynomial long division ofg(x)byh(x)yields polynomialsq(x)andr(x)∈ F[x]such that
g(x) =q(x)h(x) +r(x), wheredegr(x)<degh(x).
Moreover,q(x)andr(x)are unique. The polynomialq(x) is called thequotient, while r(x)is called theremainder. The remainder of the division is sometimes denotedg(x) mod
h(x), and the quotient is sometimes denotedg(x) divh(x)(cf. Definition 2.82).
2.192 Example (polynomial division) Consider the polynomialsg(x) =x6+x5+x3+x2+x+1 andh(x) =x4+x3+ 1inZ2[x]. Polynomial long division ofg(x)byh(x)yields
g(x) =x2h(x) + (x3+x+ 1).
Henceg(x) modh(x) =x3+x+ 1andg(x) divh(x) =x2.
2.193 Definition Ifg(x), h(x)∈F[x]thenh(x)dividesg(x), writtenh(x)|g(x), ifg(x) mod h(x) = 0.
Letf(x)be a fixed polynomial inF[x]. As with the integers (Definition 2.110), one can define congruences of polynomials inF[x]based on division byf(x).
2.194 Definition Ifg(x), h(x)∈F[x], theng(x)is said to becongruent toh(x)modulof(x) iff(x)dividesg(x)−h(x). This is denoted byg(x)≡h(x) (modf(x)).
2.195 Fact (properties of congruences) For allg(x), h(x), g1(x), h1(x), s(x) ∈ F[x], the fol-lowing are true.
(i) g(x) ≡ h(x) (modf(x))if and only ifg(x)andh(x)leave the same remainder upon division byf(x).
(ii) (reflexivity)g(x)≡g(x) (mod f(x)).
(iii) (symmetry) Ifg(x)≡h(x) (modf(x)), thenh(x)≡g(x) (modf(x)).
(iv) (transitivity) Ifg(x)≡h(x) (modf(x))andh(x)≡s(x) (modf(x)), then g(x)≡s(x) (modf(x)).
(v) Ifg(x)≡g1(x) (modf(x))andh(x)≡h1(x) (modf(x)), theng(x) +h(x)≡ g1(x) +h1(x) (modf(x))andg(x)h(x)≡g1(x)h1(x) (modf(x)).
Letf(x)be a fixed polynomial inF[x]. Theequivalence classof a polynomialg(x)∈ F[x]is the set of all polynomials inF[x]congruent tog(x)modulof(x). From properties (ii), (iii), and (iv) above, it can be seen that the relation of congruence modulof(x) par-titionsF[x]into equivalence classes. Ifg(x) ∈ F[x], then long division byf(x)yields unique polynomialsq(x), r(x)∈F[x]such thatg(x) =q(x)f(x) +r(x), wheredegr(x)
<degf(x). Hence every polynomialg(x)is congruent modulof(x)to a unique polyno-mial of degree less thandegf(x). The polynomialr(x)will be used as representative of the equivalence class of polynomials containingg(x).
2.196 Definition F[x]/(f(x))denotes the set of (equivalence classes of) polynomials inF[x]
of degree less thann= degf(x). Addition and multiplication are performed modulof(x).
2.197 Fact F[x]/(f(x))is a commutative ring.
2.198 Fact Iff(x)is irreducible overF, thenF[x]/(f(x))is a field.
2.5.5 Vector spaces
2.199 Definition Avector spaceV over a fieldF is an abelian group(V,+), together with a multiplication operation•:F×V −→V (usually denoted by juxtaposition) such that for alla, b∈Fandv, w∈V, the following axioms are satisfied.
(i) a(v+w) =av+aw.
(ii) (a+b)v=av+bv.
(iii) (ab)v=a(bv).
(iv) 1v=v.
The elements ofV are calledvectors, while the elements ofFare calledscalars. The group operation+is calledvector addition, while the multiplication operation is calledscalar multiplication.
2.200 Definition LetV be a vector space over a fieldF. AsubspaceofVis an additive subgroup U ofV which is closed under scalar multiplication, i.e.,av∈Ufor alla∈Fandv∈U. 2.201 Fact A subspace of a vector space is also a vector space.
2.202 Definition LetS={v1, v2, . . . , vn}be a finite subset of a vector spaceV over a fieldF. (i) Alinear combinationofSis an expression of the forma1v1+a2v2+· · ·+anvn,
where eachai∈F.
(ii) ThespanofS, denotedS, is the set of all linear combinations ofS. The span ofS is a subspace ofV.
(iii) IfUis a subspace ofV, thenSis said tospanU ifS=U.
(iv) The setSislinearly dependentoverFif there exist scalarsa1, a2, . . . , an, not all zero, such thata1v1+a2v2+· · ·+anvn = 0. If no such scalars exist, thenSis linearly independentoverF.
(v) A linearly independent set of vectors that spansV is called abasisforV. 2.203 Fact LetV be a vector space.
(i) IfV has a finite spanning set, then it has a basis.
(ii) IfV has a basis, then in fact all bases have the same number of elements.
2.204 Definition If a vector spaceV has a basis, then the number of elements in a basis is called thedimensionofV, denoteddimV.
2.205 Example IfFis any field, then then-fold Cartesian productV =F×F× · · · ×Fis a vector space overF of dimensionn. Thestandard basisforV is{e1, e2, . . . , en}, where eiis a vector with a1in theithcoordinate and0’s elsewhere.
2.206 Definition LetE be an extension field ofF. ThenEcan be viewed as a vector space over the subfieldF, where vector addition and scalar multiplication are simply the field operations of addition and multiplication inE. The dimension of this vector space is called thedegreeofEoverF, and denoted by[E:F]. If this degree is finite, thenEis called a finite extensionofF.
2.207 Fact LetF,E, andLbe fields. IfLis a finite extension ofEandEis a finite extension ofF, thenLis also a finite extension ofFand
[L:F] = [L:E][E:F].