Abstract Algebra
Galois Theory: Introduction
Mathematicians spent 300 years searching for a formula for the roots of degree-5 equations. A 20-year-old Galois proved no such formula exists - and died the next day. His idea: instead of formulas, study the symmetry of the roots. That symmetry is a group. And the group's structure answers the question of solvability.
- Galois theory underpins the proof that angle trisection and cube duplication are impossible with compass and straightedge
- In number theory: Galois groups over Q describe symmetries of l-adic representations, a central object of the Langlands program
Предварительные знания
Splitting Fields
The **splitting field** of f ∈ F[x] is the smallest extension L ⊇ F in which f factors completely into linear factors: f(x) = a(x − α₁)(x − α₂)···(x − αₙ), αᵢ ∈ L. The splitting field is unique up to isomorphism over F. It is constructed iteratively: adjoin roots one by one - F ⊂ F(α₁) ⊂ F(α₁,α₂) ⊂ ... = L.
**Évariste Galois (1811-1832)** was a French mathematician who created Galois theory at around age 20. On the night before a duel (over a woman, or politics - historians disagree), he wrote down his mathematical discoveries, sensing his end was near. He wrote to his brother: 'I don't have time.' The next day he was fatally wounded. His manuscripts were published only 14 years after his death. Today, Galois theory is the foundation of modern algebra.
A **normal extension** L/F is one that is the splitting field of some polynomial over F. Equivalently: every irreducible polynomial over F that has one root in L splits completely in L. A **separable extension** is one where all minimal polynomials have only simple roots (automatic in characteristic 0).
What is the splitting field of f = x² + 1 over R (the real numbers)?
The Galois Group
Let L/F be a field extension. The **Galois group** Gal(L/F) is the group of all automorphisms of L that fix F: Gal(L/F) = {σ: L → L | σ is an automorphism, σ(a) = a for all a ∈ F}. An automorphism σ ∈ Gal(L/F) permutes the roots of irreducible polynomials over F: if f(α) = 0, then f(σ(α)) = 0. So Gal(L/F) acts on the roots of every f ∈ F[x].
A **Galois extension** is one that is both normal and separable. Not every field extension is Galois. For example, Q(∛2)/Q is not Galois: ω ∉ Q(∛2), so not all roots of x³−2 lie in Q(∛2), making it non-normal. For non-Galois extensions |Gal(L/F)| < [L:F].
**The Galois group as a permutation group.** For f ∈ F[x] of degree n with roots α₁,...,αₙ in the splitting field L: each σ ∈ Gal(L/F) acts on {α₁,...,αₙ} as a permutation. This gives an embedding Gal(L/F) ↪ Sₙ. For a 'generic' polynomial of degree n, the Galois group is all of Sₙ.
What is the order of the Galois group Gal(Q(√2, √3)/Q)?
The Fundamental Theorem of Galois Theory
**Fundamental Theorem of Galois Theory:** Let L/F be a finite Galois extension with group G = Gal(L/F). Then there is a bijective correspondence (anti-isomorphism of lattices): {Intermediate fields F ⊆ K ⊆ L} ↔ {Subgroups H ≤ G} Correspondence: K ↦ Gal(L/K) = {σ ∈ G | σ(k) = k ∀k ∈ K}, H ↦ L^H = {x ∈ L | σ(x) = x ∀σ ∈ H}. Moreover: [L:K] = |Gal(L/K)|, [K:F] = [G:Gal(L/K)], and K/F is normal ⟺ Gal(L/K) ◁ G.
**Connection to solvability of equations.** An equation f = 0 is solvable by radicals ⟺ the Galois group of f is solvable. Quadratics: Gal ⊆ Z₂ (solvable → take √). Cubics: Gal ⊆ S₃ (solvable → Cardano's formula). Degree ≥ 5: Sₙ is not solvable for n≥5 → the 'generic' degree-5 equation has no radical formula (Abel-Ruffini theorem).
**Why is this revolutionary?** Before Galois, mathematicians tried to find formulas for roots of degree-5 equations (analogous to Cardano's formula for cubics). Galois reframed the question: instead of 'find the roots' - 'what is the symmetry of the roots?' The answer is a group. This shift from computation to structure defined 20th-century mathematics: category theory, homological algebra, modern geometry - all think in terms of structures, not formulas.
The Galois group is simply the group of permutations of the roots of a polynomial
The Galois group is the group of automorphisms of the splitting field that fix the base field. The action on roots is a consequence, not the definition.
Defining it via field automorphisms enables the fundamental correspondence with intermediate fields - the most powerful result of the theory.
A Galois extension L/Q has Galois group G = S₃. How many intermediate fields lie between Q and L?
Key Ideas
- Splitting field of f: the smallest L ⊇ F where f factors completely
- Gal(L/F) = group of automorphisms of L fixing F; |Gal(L/F)| = [L:F] for Galois extensions
- Fundamental theorem: intermediate fields ↔ subgroups of Gal(L/F) (anti-isomorphism)
- K/F is normal ⟺ Gal(L/K) ◁ Gal(L/F)
- f solvable by radicals ⟺ Gal(f) is solvable
Further Paths
Galois theory is the pinnacle of classical algebra. It unites group theory, field theory, and number theory into a single whole.
- Field Extensions — Degree of extension and minimal polynomials - the technical foundation of Galois theory
- Solvable Groups — Solvability of the Galois group ↔ solvability by radicals - the central theorem
Вопросы для размышления
- Build the Galois correspondence diagram for Q(∛2, ω)/Q (G ≅ S₃). Find all 6 intermediate fields.
- Why must a field automorphism fixing F necessarily permute the roots of any irreducible polynomial over F?
- How does Galois theory explain the existence of Cardano's formula for cubics but the absence of an analogous formula for degree-5 equations?