Abstract Algebra - Groups, Rings, and Fields

Table of Contents

1. Groups
2. Rings
3. Fields
4. Vector Spaces
5. Modules
6. Galois Theory
7. Category Theory

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Groups

Definition

Group (G, ·): Set G with operation · satisfying:

1. Associativity: (a · b) · c = a · (b · c)
2. Identity: ∃ e ∈ G such that e · a = a · e = a
3. Inverse: For each a ∈ G, ∃ a^(-1) ∈ G with a · a^(-1) = a^(-1) · a = e

Abelian: a · b = b · a (commutative)

Order of group: |G| = number of elements

Examples

ℤ under addition: (ℤ, +) abelian, infinite

Symmetric group S_n: Permutations of n elements, |S_n| = n!

General linear group GL_n(F): Invertible n×n matrices over field F

Dihedral group D_n: Symmetries of regular n-gon, |D_n| = 2n

Subgroups

H ⊆ G is subgroup if closed under operation and inverses

Notation: H ≤ G

Test: For all a,b ∈ H: a · b ∈ H, a^(-1) ∈ H

Cyclic Groups

Generated by single element: G = ⟨g⟩ = {g^n : n ∈ ℤ}

Finite cyclic: G = {e, g, …, g^(n-1)} where g^n = e

Any two finite cyclic groups of same order isomorphic to ℤ/nℤ

Lagrange’s Theorem

If H subgroup of finite group G: $|H| \text{ divides } |G|$

Order of element divides order of group

Consequence: a^(|G|) = e

Normal Subgroups

H is normal (H ⊲ G) if for all g ∈ G: gH = Hg

Equivalently: ghg^(-1) ∈ H for all g ∈ G, h ∈ H

Kernel of homomorphism is normal

Quotient Groups

G/H defined when H ⊲ G

Elements: Cosets gH

Operation: (aH)(bH) = (ab)H

|G/H| = |G|/|H|

Group Homomorphisms

φ: G → H homomorphism if φ(ab) = φ(a)φ(b)

Properties:

• φ(e) = e
• φ(a^(-1)) = φ(a)^(-1)
• ker(φ) = {a : φ(a) = e} is subgroup (normal)

Isomorphism: Bijective homomorphism

Fundamental theorem: If φ: G → H, then G/ker(φ) ≅ im(φ)

Permutation Groups

Cycle notation: (a₁ a₂ … aₖ)

Disjoint cycles commute

Every permutation product of disjoint cycles

Parity: Even (product of even number of transpositions) or odd

Alternating group A_n: Even permutations, |A_n| = n!/2

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Rings

Definition

Ring (R, +, ·): Set R with two operations such that:

1. (R, +) is abelian group
2. Multiplication associative: (ab)c = a(bc)
3. Distributive: a(b+c) = ab+ac, (a+b)c = ac+bc

Commutative ring: ab = ba

Ring with unity: Has multiplicative identity 1

Integral domain: Commutative, 1 ≠ 0, no zero divisors (ab=0 ⟹ a=0 or b=0)

Field: Commutative ring where every nonzero element has multiplicative inverse

Examples

ℤ: Integers (integral domain)

ℝ[x]: Polynomials (commutative ring)

M_n(F): n×n matrices over field (not commutative)

ℤ/nℤ: Integers modulo n (field if n prime)

Ideals

I ⊆ R is ideal if:

• (I, +) subgroup
• For all r ∈ R, a ∈ I: ra ∈ I and ar ∈ I

Principal ideal: (a) = {ra : r ∈ R}

Maximal ideal: No larger proper ideal

Prime ideal: If ab ∈ I, then a ∈ I or b ∈ I

Quotient Rings

R/I for ideal I

Elements: Cosets r + I

Operations: (a+I) + (b+I) = (a+b)+I (a+I)(b+I) = ab + I

R/I integral domain ⟺ I prime R/I field ⟺ I maximal

Euclidean Domains

Integral domain with Euclidean function d: R - {0} → ℕ such that:

1. For division: a = qb + r with r=0 or d(r) < d(b)
2. For all a,b ≠ 0: d(a) ≤ d(ab)

Examples: ℤ, F[x] where F is a field

Every Euclidean domain is PID

Principal Ideal Domains (PID)

Integral domain where every ideal is principal

Examples: ℤ, F[x]

PID has unique factorization

Factorization

Irreducible: Cannot be written as product of nontrivial factors

Prime: p|ab ⟹ p|a or p|b

In integral domain: Prime ⟹ irreducible In PID: Irreducible ⟺ Prime

UFD (Unique Factorization Domain): Every element factors uniquely

Every PID is UFD

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Fields

Definition

Field F: Commutative ring with 1 ≠ 0 where every nonzero element has inverse

Subfield: K ⊆ F is subfield if closed under operations

Field Extensions

F extension of K: F ⊇ K (K subfield of F)

Degree: [F : K] = dimension of F as vector space over K

Tower: If L/K, F/L, then [F : K] = [F : L][L : K]

Algebraic Extensions

α algebraic over K: Root of nonzero polynomial over K

Minimal polynomial: Monic polynomial of smallest degree with α as root

Degree of α: deg(α) = degree of minimal polynomial

F/K algebraic: Every element of F is algebraic over K

Splitting Fields

F is splitting field for polynomial over K if F contains all roots

For f ∈ K[x]: Smallest extension where f splits completely

Finite Fields

Every finite field has pⁿ elements where p prime

GF(pⁿ) or 𝔽_{pⁿ}: Unique field of pⁿ elements

F_p: Prime field ℤ/pℤ

Construction: GF(pⁿ) = splitting field of x^(pⁿ) - x over F_p

Multiplicative group: GF(pⁿ)* ≅ ℤ/(pⁿ-1)ℤ

Cyclic: Always has primitive element

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Vector Spaces

See Linear Algebra for full treatment.

Vector space over field F: Abelian group with scalar multiplication

Linear independence, basis, dimension

Fields can be viewed as vector spaces over subfields

Linear Transformations

T: V → W is linear if:

• T(u + v) = T(u) + T(v)
• T(cv) = cT(v)

Matrix representation via choice of bases

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Modules

Definition

R-module M: Like vector space but scalars from ring R

Left module: Operation R × M → M Right module: Operation M × R → M

Vector space = module over field

Structure Theorems

For PID R:

M finitely generated ⟺ M ≅ Rⁿ ⊕ R/(a₁) ⊕ … ⊕ R/(aₖ)

where a_i | a_{i+1}

Invariant factors a_i determined uniquely

Free Modules

Free module: Has basis

Rank: Size of basis (well-defined for commutative R)

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Galois Theory

Galois Groups

Automorphism of field F: Bijective map φ: F → F preserving + and ·

Gal(F/K): Group of automorphisms fixing K pointwise

|Gal(F/K)| ≤ [F : K]

Normal Extensions

F/K normal if every irreducible polynomial in K[x] with a root in F splits

Fixed field: K^G = {α : σ(α) = α for all σ ∈ G}

Fundamental Theorem of Galois Theory

For Galois extension F/K:

Correspondence between:

• Subgroups H ≤ Gal(F/K)
• Intermediate fields E with K ⊆ E ⊆ F

Bijection: H ↔ K^H (fixed field of H)

Properties:

• [F : E] = |H|
• E is Galois over K ⟺ H is normal in Gal(F/K)
• If H normal, then Gal(E/K) ≅ Gal(F/K)/H

Solvability by Radicals

Polynomial f(x) solvable by radicals if roots can be expressed using +, -, ·, /, and radicals

Galois group is solvable (has chain with abelian quotients)

Result: S₅ not solvable → Degree 5+ polynomials not generally solvable by radicals

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Category Theory

Categories

Category C: Objects and morphisms between them

Composition: For f: A → B, g: B → C: g ∘ f: A → C

Identity: 1_A: A → A

Associativity: (h ∘ g) ∘ f = h ∘ (g ∘ f)

Functors

F: C → D: Maps objects and morphisms preserving:

• Identities: F(1_A) = 1_{F(A)}
• Composition: F(g ∘ f) = F(g) ∘ F(f)

Covariant: F(f: A → B) = F(A) → F(B) Contravariant: F(f: A → B) = F(B) → F(A)

Natural Transformations

η: F → G: Family of morphisms η_A: F(A) → G(A) making square commutative

Universal Properties

Initial/terminal objects Products/coproducts Pullbacks/pushouts

Applications

Homological algebra Topology (homotopy theory) Algebraic topology Model categories

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Next: Fourier Analysis or Tensor Analysis

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Last updated: Comprehensive abstract algebra reference covering groups, rings, fields, and category theory.