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 −1 Hg = 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 pn m 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,
|HK| =
|H| · |K|
.
|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) (g1 g2 )(x) = g1 (g2 x), ∀ 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.
Dihedral Group The nth 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

1 a
0 1
0 0
in H(n) and the inverses are given by
−1 

c
1 −a −ab − c
b  = 0 1
−b  .
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:
|GLn (F )| = (q n − 1)(q n − q)(q n − q 2 ) · · · (q n − q n−1 ).
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
(h, k)(h0 , k 0 ) = (hαk (h0 ), kk 0 ).
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.
Let α : K → Aut(H) be given by αk (h) = khk −1 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 domain 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 preserves 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
Eisenstein’s Criterion The polynomial p(x) = an xn + an−1 xn−1 + . . . + a1 x + 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
p2 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,
• F [α] = {b0 + b1 α + . . . + bm αm | b0 , . . . bm ∈ F, m ≥ 0}
→ {g(α)|g(x) ∈ F [x]}
Ring, sometimes a field ; smallest subring of E containing α and F .
b0 + b1 α + . . . + bm αm • F (α) =
b , . . . , bm ∈ F, c0 , . . . cr ∈ F, m, r ≥ 0
r 0
c0 + c1 α + . . . + cr α 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
nth Cyclotomic Polynomial
The nth cyclotomic polynomial, Φn (x), is defined to be
Y k
Φn (x) =
x − e2πi n .
1≤k≤n
gcd(k,n)=1
k
e2π 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 k n−k
n
(x + y) =
x y .
k
k=0
Elementary Symmetric Polynomials For k ≥ 0, we define the elementary symmetric polynomial in n variables as
X
ek =
xj 1 · · · xj k ,
1≤j1 <j2 <...<jk ≤n
so that ek = 0 if k > n.