Galois Theory: An Overview

Two thousand years of mathematicians searched for a quintic formula. Évariste Galois proved at age 19 that it cannot exist-and in doing so created group theory, which now underlies cryptography, error correction, and even type theory. Galois theory is one of the most beautiful chapters in all of mathematics.

• **LFSR in stream ciphers:** pseudo-random generators in A5/1 (GSM) and Trivium use primitive polynomials over GF(2); primitivity is a question of Galois theory
• **AES structure:** GF(2⁸) is the Galois extension GF(2⁸)/GF(2) of degree 8; the Frobenius automorphism x ↦ x² appears explicitly in the cipher
• **Algebraic geometry:** algebraic curves over finite fields underlie elliptic curve cryptography; their point-counting uses Galois theory

Предварительные знания

• Rings and Fields

Field Extensions

A field extension L/K means L contains K as a subfield. The degree [L:K] = dim_K(L) is the dimension of L as a vector space over K. Examples: [ℂ:ℝ] = 2 (basis {1, i}); [ℝ:ℚ] = ∞. If f(x) ∈ K[x] is irreducible and α is a root in L, then K(α) = {a₀ + a₁α + … + a_{n-1}αⁿ⁻¹ : aᵢ ∈ K} with [K(α):K] = deg(f).

**Splitting field:** the smallest extension of K in which f(x) factors completely into linear factors. For f(x) = xⁿ − 1, the splitting field over ℚ is ℚ(ωₙ) where ωₙ = e^(2πi/n)-the cyclotomic extension, directly connected to the DFT and FFT.

The Galois Group

The Galois group Gal(L/K) is the group of all K-automorphisms of L-bijective maps φ: L → L that preserve all field operations and fix K pointwise. For ℚ(√2)/ℚ: the automorphisms are the identity and √2 ↦ −√2. So Gal(ℚ(√2)/ℚ) ≅ ℤ/2ℤ.

**Galois extension:** a normal and separable extension L/K. For these, |Gal(L/K)| = [L:K]. All extensions of finite fields GF(pⁿ)/GF(p) are Galois with Gal ≅ ℤ/nℤ, generated by the Frobenius automorphism x ↦ xᵖ.

Fundamental Theorem of Galois Theory

The Fundamental Theorem of Galois Theory establishes a bijection between intermediate fields K ⊆ M ⊆ L and subgroups {e} ≤ H ≤ Gal(L/K): the field M corresponds to Gal(L/M), and the subgroup H corresponds to the fixed field L^H = {x ∈ L : σ(x) = x for all σ ∈ H}. The correspondence reverses inclusion.

**Normal subgroups ↔ normal extensions:** if H ◁ Gal(L/K), the corresponding intermediate field M/K is a normal extension, and Gal(M/K) ≅ Gal(L/K)/H. Normal subgroups correspond to extensions that can themselves be analyzed by Galois theory.

Solvability by Radicals

A polynomial f(x) ∈ ℚ[x] is solvable by radicals if and only if its Galois group Gal(f) is a solvable group. Sₙ is solvable for n ≤ 4 but S₅ is not (A₅ is simple, non-abelian). Therefore, a general polynomial of degree ≥ 5 has no formula in radicals (Abel-Ruffini theorem, 1826).

LFSR (Linear Feedback Shift Register) in stream ciphers (A5/1 in GSM, Trivium) uses primitive polynomials over GF(2). A polynomial is primitive iff it generates GF(2ⁿ) as a field extension-a direct Galois theory concept.

Key Ideas

• **Field extension L/K:** degree [L:K] = dim_K(L); tower law: [L:K] = [L:M]·[M:K]
• **Galois group Gal(L/K):** K-automorphisms of L; |Gal| = [L:K] for Galois extensions
• **Fundamental theorem:** intermediate fields ↔ subgroups (inclusion reversed); normal subgroups ↔ normal extensions
• **Solvability:** f solvable by radicals ↔ Gal(f) solvable; S₅ not solvable → no formula for degree ≥ 5 (Abel-Ruffini)

Related Topics

Galois theory unifies groups, fields, and polynomials:

• Groups — The Galois group is a specific group; solvability of the group is the algebraic criterion
• Rings and Fields — Field extensions are the central object of Galois theory
• Polynomials — The splitting field of a polynomial is Galois theory's starting point

Вопросы для размышления

• Galois created group theory to solve a problem about polynomials. How often in mathematics does an abstract theory created for one specific problem find applications in a completely different domain?
• The Frobenius automorphism x ↦ xᵖ generates Gal(GF(pⁿ)/GF(p)). How is it used in elliptic curve point-counting algorithms like Schoof's algorithm?
• Galois theory also explains why you cannot trisect an angle with compass and straightedge. How does this connect to the degree of field extensions?

Связанные уроки

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