ALGEBRA PRELIM - DEFINITIONS AND THEOREMS

HARINI CHANDRAMOULI

Group Theory Group A group is a set, G, together with an operation · that must satisfy the following group axioms: • Closure. ∀ a, b ∈ G, a · b ∈ G. • Associativity. ∀ a, b, c ∈ G,(a · b) · c = a · (b · c). • Identity Element. ∃ e ∈ G such that ∀a ∈ G, e · a = a · e = a. Such an element is unique. • Inverse Element. ∀ a ∈ G, ∃ a−1 ∈ G such that a · a−1 = a−1 · a = e.

p-Sylow Subgroup Let pe be the largest power of p dividing the order of G.A p-Sylow subgroup (if it exists) is a subgroup of G of order pe.

Sylow Theorems

Theorem 1. For any prime factor p with multiplicity n of the order of a finite group G, there exists a p-Sylow subgroup of G, of order pn.

Theorem 2. Give a finite group G and a prime number p, all p-Sylow subgroups of G are conjugate to each other, i.e. if H and K are p-Sylow subgroups of G, then there exists an element g ∈ G with g−1Hg = K.

Theorem 3. Let p be a prime factor with multiplicity n of the order of a finite group G, so that the order of G can be written as pnm where n > 0 and p does not divide m. Let

np be the number of p-Sylow subgroups of G. Then the following hold: (1) np divides m, which is the index of the p-Sylow subgroup in G (2) np ≡ 1 (mod p) (3) np = |G : NG(P )| where P is any p-Sylow subgroup of G and NG denotes the normalizer

Lagrange’s Theorem For any finite group G, the order of every subgroup H of G divides the order of G. Note: Every group of prime order is cyclic.

Theorem Every group of even order contains an element of order 2. 1 2 HARINI CHANDRAMOULI

Fundamental Theorem of Finite Abelian Groups Every finite abelian group is an internal group direct product of cyclic groups whose orders are prime powers.

Theorem Let G be a group and p be a prime number. If |G| = p2, then G is abelian.

Theorem Let H and K be subgroups of G. Then, |H| · |K| |HK| = . |H ∩ K|

Index of a Subgroup The index of a subgroup H in a group G is the “relative size” of H in G; equivalently, the number of “copies” (cosets) of H that fill up G. → If N is a normal subgroup of G, then the index of N in G is also equal to the order of the quotient group G/N, since this is defined in terms of a group structure on the set of cosets of N in G.

Action of G on X Let X be a set and G a group. An action of G on X is a map ∗ : G × X → X such that (1) ex = x, ∀ x ∈ X.

(2) (g1g2)(x) = g1(g2x), ∀ x ∈ X and ∀ g1, g2 ∈ G.

Theorem Let p be the smallest prime dividing the order of a group G. If H is a subgroup of G with index p, then H is normal.

Theorem Let G be a group and H, K be subgroups of G. Note that the product set HK is defined as HK := {hk | h ∈ H, k ∈ K} . If H is normal in G, then HK is a subgroup of G. Moreover, if both H and K are normal in G, then HK is a normal subgroup of G.

Quotient Group Let H be a normal subgroup of G. Then the cosets of H form a group G/H under the binary operation (aH)(bH) = (ab)H. The group G/H is the quotient group or factor group of G by H.

th Dihedral Group The n dihedral group, D2n, is the group of symmetries of the regular n-gon. → A regular polygon with n sides has 2n different symmetries: n rotational symmetries, and n reflection symmetries. The associated rotations and reflections makeup the dihedral

group D2n. → The composition of two symmetries of a regular polygon is again a symmetry of this object, giving us the algebraic structure of a finite group.

→ The dihedral group with two elements, D2, and the dihedral group with four elements, D4, are abelian. For all other n, D2n is not abelian. ALGEBRA PRELIM - DEFINITIONS AND THEOREMS 3

Non-Abelian Group of order n3: Let n ∈ N, n ≥ 2. A classical example of a non-abelian group of order n3 is the Heisenberg Group:    1 a c   H(n) = 0 1 b a, b, c ∈ Z/nZ .

 0 0 1  We see that the identity matrix is in H(n) and the inverses are given by −1 1 a c 1 −a −ab − c 0 1 b = 0 1 −b  . 0 0 1 0 0 1

Theorem GLn(F ) is the general linear group of degree n over a field F with n×n invertible matricies, together with the operation of ordinary matrix multiplication. Let F be a finite

field with q elements. Then the order of GLn(F ) is given as follows: n n n 2 n n−1 |GLn(F )| = (q − 1)(q − q)(q − q ) ··· (q − q ).

Semidirect Product Let H and K be subgroups of a group G such that H C G. Suppose that α : K → Aut(H) is a homomorphism between the group K and the automorphism group of the group H. Then the semidirect product of H and K determined by α, denoted by H oα K, is the set H × K equipped with the binary operation 0 0 0 0 (h, k)(h , k ) = (hαk(h ), kk ).

Theorem The semidirect product H oα K is a group under binary operation given above.

Theorem Suppose that H and K are subgroups of G satisfying (1) H ∩ K = {e}. (2) H C G. (3) HK = G. −1 Let α : K → Aut(H) be given by αk(h) = khk be the automorphism of H given by the restriction of the conjugation operator to K, acting on H. Then the map H oα K → G given by (h, k) 7→ hk is an isomorphism.

Theorem Let p be a prime number. Then, ∼ × Aut (Z/pZ) = (Z/pZ) .

First Isomorphism Theorem Let G and H be groups, let ϕ : G → H be a homomorphism. Then: (1) The kernel of ϕ is a normal subgroup of G. (2) The image of ϕ is a subgroup of H, and (3) The image of ϕ is isomorphic to the quotient group G/ ker(ϕ). In particular, if ϕ is surjective, then H is isomorphic to G/ ker(ϕ). 4 HARINI CHANDRAMOULI

Rings and Fields Ring A ring is a set, R, equipped with binary operations + and · satisfying the following ring axioms:

• R is an abelian group under addition. That is, (R, +) satisfies the group axioms and in addition, a + b = b + a holds ∀ a, b ∈ R. • R has a multiplicative identity and multiplication is associative. That is, (a · b) · c = a · (b · c) holds ∀ a, b, c ∈ R and ∀ a ∈ R, ∃ 1 ∈ R such that 1 · a = a · 1 = a. • Multiplication is distributive with respect to addition. That is, ∀a, b, c ∈ R, a·(b+c) = (a · b) + (a · c) holds (left distributivity) and (b + c) · a = (b · a) + (c · a) holds (right distributivity).

Field A field is a set, F , equipped with two operations, + and ·, satisfying the following axioms:

• Closure of F under addition and multiplication. That is ∀ a, b ∈ F , a + b ∈ F and a · b ∈ F . • Associativity of addition and multiplication. That is, ∀ a, b, c ∈ F , a + (b + c) = (a + b) + c and a · (b · c) = (a · b) · c. • Commutativity of addition and multiplication. That is, ∀ a, b ∈ F , a + b = b + a and a · b = b · a. • Existence of additive and multiplicative identity elements. That is, ∀ a ∈ F , ∃ 0 ∈ F and ∃ 1 ∈ F such that 0 + a = a and 1 · a = a. • Existence of additive and multiplicative inverses. That is, ∀ a ∈ F , ∃ − a ∈ F and ∃ a−1 ∈ F such that a + (−a) = 0 and a · a−1 = 1. • Distributivity of multiplication over addition. That is, ∀ a, b, c ∈ F , a · (b + c) = (a · b) + (a · c).

Integral Domain An integral domain is a commutative ring with multiplicative identity and no divisors of 0.

Unique Factorization Domain A unique factorization domain (UFD) is an integral do- main in which every nonzero, noninvertible element has a unique factorization, that is, a decomposition as the product of prime elements or irreducible elements.

Euclidean Domain Let R be an integral domain. A Euclidean function on R is a function f from R\{0} to the non-negative integers such that if a, b ∈ R and b 6= 0, then there are q, r ∈ R such that a = bq + r and either r = 0 or f(r) < f(b). A Euclidean domain is an integral domain which can be endowed with at least one Euclidean function. → Note that a particular Euclidean function f is NOT part of the structure of a Euclidean domain. In general, a Euclidean domain will admit many different Euclidean functions. → Examples: Z - the ring of integers, K[x] - the ring of polynomials over a field K ALGEBRA PRELIM - DEFINITIONS AND THEOREMS 5

→ Every ideal in a Euclidean domain is principal. → Every Euclidean domain is a UFD.

Ideal Let (R, +, ·) be a ring and (R, +) be the additive subgroup. Subset I is called an ideal if (I, +) is a subgroup of (R, +) and ∀ x ∈ I, and ∀ r ∈ R, x · r ∈ I and r · x ∈ I.

Principal Ideal Let R be a commutative ring, and let a ∈ R. I is a principal ideal if I = hai = {ra | r ∈ R}, that is, I can be generated by a.

Maximal Ideal Let R be a ring and M ⊂ R, M an ideal such that M is not equal to R. M is a maximal ideal if @ I ⊂ R, I an ideal, such that M ⊂ I ⊂ R.

Prime Ideal Let R be a ring, P ⊂ R, P an ideal. P is a prime ideal if the following holds: if ab ∈ P , then a ∈ P or b ∈ P .

Principal Ideal Domain A principal ideal domain (PID) is an integral domain in which every ideal is principal. → ex. F - a field, then in F [x] every ideal is principal. Z is also a PID.

Theorem M is a maximal ideal iff R/M is a field.

Theorem Let F be a field, N an ideal of F [x]. TFAE: (a) N = hf(x)i is maximal. (b) N is prime. (c) f(x) is irreducible.

Commutative Ring with Unity and a Non-Principal Ideal: Consider the ring Z[x]. Clearly this is commutative and 1 ∈ Z[x]. Consider now the ideal I = h2, xi. This ideal can be shown to be non-principal. 6 HARINI CHANDRAMOULI

Modules Module Let R be a ring. An R-module M is an abelian group M (with operation written additively) and a multiplication R × M → M written r × m → r · m = rm such that for r, s ∈ M and x, y ∈ M (1) r · (x + y) = r · x + r · y (distributivity) (2) (r + s) · x = r · x + s · x (distributivity) (3) (r · s) · x = r · (s · x) (associativity)

We specifically do not universally require that 1R · x = x, ∀ x ∈ M in an R-module M when the ring R contains a unit 1R. Nevertheless, on many occasions, we do require this, but, therefore, must say so explicitly to be clear. → A module over a ring is a generalization of the notion of a vector space over a field, wherein the corresponding scalars are the elements of an arbitrary given ring. → A module is an additive abelian group, a product is defined between elements of the ring and elements of the module that is distributive over the addition operation of each parameter and is compatible with ring multiplication. → If K is a field, then a vector space over K and a K-module are identical. → If K is a field, and K[x] a polynomial ring, then a K[x]-module M is a K-module with an additional action of x on M that commutes with the action of K on M. In other words, a K[x]-module is a K-vectorspace M combined with a linear map from M to M. → The concept of a Z-module agrees with the notion of an abelian group. That is, every abelian group is a module over the ring of integers Z in a unique way. → If R is any ring and I is any ideal in R, then I is a module over R.

Finitely-Generated Modules Let R be a principal ideal domain. An R-module M is finitely

generated if there are finitely-many m1, . . . , mn ∈ M such that every element m ∈ M is expressible in at least one way as

m = r1 · m1 + . . . rn · mn with ri ∈ R. → A basic construction of new R-modules from old is as direct sums. Given R-modules

M1,...Mn, the direct sum R-module M1⊕· · ·⊕Mn is the collection of n-tuples (m1, . . . , mn) with mi ∈ Mi with component-wise addition and multiplication by elements r ∈ R by r · (m1, . . . , mn) = (rm1, . . . rmn). ALGEBRA PRELIM - DEFINITIONS AND THEOREMS 7

Structure Theorem Let M be a finitely-generated module over a PID R. Then there are uniquely determined ideals

I1 ⊃ I2 ⊃ ... ⊃ It such that

M ≈ R/I1 ⊕ R/I2 ⊕ · · · R/It.

The ideals Ii are the elementary divisors of M, and this expression is the elementary divisor form of M.

Structure Theorem for Z-modules Let M be a finitely-generated Z-module (that is, a finitely-generated abelian group). Then there are uniquely determined non-negative integers

d1, . . . , dn such that

d1 d2 ··· dn and M ≈ Z/d1 ⊕ Z/d2 ⊕ · · · ⊕ Z/dn.

Module Homomorphism A module homomorphism is a function between modules that pre- serves module structures. Explicitly, if M and N are modules over a ring R, then a function f : M → N is called a module homomorphism or a R-linear map if for any x, y ∈ M and r ∈ R, (1) f(x + y) = f(x) + f(y) (2) f(rx) = rf(x) 8 HARINI CHANDRAMOULI

Reducibility

n n−1 Eisenstein’s Criterion The polynomial p(x) = anx + an−1x + ... + a1x + a0, where ai ∈ Z, ∀ i = 0, . . . , n and an 6= 0 (that is, p(x) has degree n) is irreducible if some prime number p divides all coefficients a0, . . . , an−1 but not the leading coefficients an, and moreover 2 p does not divide the constant term a0.

Gauss’s Lemma Let R be a unique factorization domain and F its field of fractions. If p(x) ∈ R[x] is irreducible, then p(x) is also irreducible in F [x].

Notation: F,E - fields, F ⊂ E, and α ∈ E. Then, m • F [α] = {b0 + b1α + ... + bmα | b0, . . . bm ∈ F, m ≥ 0} → {g(α)|g(x) ∈ F [x]} Ring, sometimes a field ; smallest subring of E containing α and F .

 m  b0 + b1α + ... + bmα • F (α) = r b0, . . . , bm ∈ F, c0, . . . cr ∈ F, m, r ≥ 0 c0 + c1α + ... + crα   g(α) → g(x), h(x) ∈ F [x] h(α) Field, smallest subfield of E containing α and F • Note that C(x) and C[x] are the same since C is algebraically closed, that is, the inverses are already there! ALGEBRA PRELIM - DEFINITIONS AND THEOREMS 9

Field Extensions and Galois Theory Degree of Extension If an extension E of a field F is of finite dimension n as a vector space over F , then E is a finite extension of degree n over F . We shall let [E : F ] denote the degree n of E over F .

Index of Extension Let E be a finite extension of a field F . The number of isomorphisms of E onto a subfield of F leaving F fixed is the index {E : F } of E over F .

Splitting Field Let F be a field with algebraic closure F . Let {fi(x) | i ∈ I} be a collection of polynomials in F [x]. A field E ≤ F is the splitting field of {fi(x) | i ∈ I} over F if E is the smallest subfield of F containing F and all the zeros in F of each of the fi(x) for i ∈ I. A field K ≤ F is a splitting field over F if it is the splitting field of some set of polynomials in F [x].

Separable Extension A finite extension E of F is a separable extension of F if {E : F } = [E : F ]. An element α of F is separable over F if F (α) is a separable extension of F . An irreducible polynomial f(x) ∈ F [x] is separable over F if every zero of f(x) in F is separable over F .

Fundamental Theorem of Galois Theory F,E - fields, F ⊂ E a Galois extension (i.e. separable and splitting field). Gal(E/F ) is the Galois group, i.e. the group of automorphisms of E fixing F . There is a one-to-one correspondence between intermediate fields, F ⊂ K ⊂ E, and subgroups of the Galois group, {e} ⊂ H ⊂ Gal(E/F ). 10 HARINI CHANDRAMOULI

Commuting Operators

Diagonalizable A linear operator T ∈ Endk(V ) on a finite-dimensional vectorspace V over a field k is diagonalizable if V has a basis consisting of eigenvectors of T .

Corollary (found in Paul Garrett’s Notes, 24.1.5) Let k be algebraically closed, and V a finite-dimensional vectorspace over k. Then there is at least one eigenvalue and (non-zero) eigenvector for any T ∈ Endk(V ).

Proposition (found in Paul Garrett’s Notes, 24.2.2) An operator T ∈ Endk(V ) with V a finite-dimensional vectorspace over the field k is diagonalizable iff the minimum polynomial f(x) of T factors into linear factors in k[x] and has no repeated factors. Further, letting Vλ be the λ-eigenspace, diagonalizability is equivalent to X V = Vλ. eigenvalues λ ALGEBRA PRELIM - DEFINITIONS AND THEOREMS 11

Miscellaneous

th th n Cyclotomic Polynomial The n cyclotomic polynomial,Φn(x), is defined to be

Y  2πi k  Φn(x) = x − e n . 1≤k≤n gcd(k,n)=1

2π k e n where gcd(k, n) = 1 are called the primitive roots of unity. → if n = p a prime, then Φp(x) is irreducible over Q and subsequently, Z. xp − 1 → = xp−1 + xp−2 + ... + 1 is the pth cyclotomic polynomial. x = 1

Binomial Theorem It is possible to expand any power of x + y into a sum of the form n X n (x + y)n = xkyn−k. k k=0

Elementary Symmetric Polynomials For k ≥ 0, we define the elementary symmetric poly- nomial in n variables as X ek = xj1 ··· xjk ,

1≤j1 n.