Galois Theory¶

Summary

At the age of 19, Évariste Galois discovered that the solvability of an equation is determined by the symmetries of its roots. His theory connects field extensions with groups – and provides the conceptual framework for Wiles' proof.

Prerequisites¶

1. The Problem of Solution Formulas¶

Everyone knows the quadratic formula:

\[ x^2 + bx + c = 0 \implies x = \frac{-b \pm \sqrt{b^2 - 4c}}{2} \]

The roots can be expressed in terms of the coefficients – using addition, subtraction, multiplication, division, and root extraction.

For equations of degree three and four, there likewise exist (more complicated) solution formulas, discovered by Cardano (1545) and Ferrari. Naturally the question arises: do such formulas also exist for degree \(5\) and higher?

The answer is no – and the justification leads directly to Galois theory.

2. Abel's Impossibility Proof¶

Niels Henrik Abel proved in 1824 that there is no general solution formula for polynomials of degree \(\geq 5\) in radicals. That is: there exist fifth-degree polynomials whose roots cannot be expressed by finitely many root extractions from the coefficients.

Abel's proof was a milestone, but left a decisive question open: which polynomials can be solved by radicals and which cannot? A specific fifth-degree polynomial may well have a solution in radicals – for instance \(x^5 - 2 = 0\) with \(x = \sqrt[5]{2}\). Abel's theorem only says that no general procedure exists.

3. Galois' Idea¶

Évariste Galois (1811–1832) solved this problem completely – with an idea decades ahead of its time. He died at 20 in a duel; the night before, he feverishly wrote down his mathematical insights.

Galois' central insight: The solvability of an equation is determined by the symmetries of its roots.

Consider a polynomial \(f \in \mathbb{Q}[x]\) with roots \(\alpha_1, \ldots, \alpha_n \in \overline{\mathbb{Q}}\). The splitting field is the smallest field containing \(\mathbb{Q}\) and all roots:

\[ L = \mathbb{Q}(\alpha_1, \ldots, \alpha_n) \]

A symmetry of this field is an automorphism \(\sigma: L \to L\) that fixes \(\mathbb{Q}\) element-wise. Every such automorphism permutes the roots \(\alpha_1, \ldots, \alpha_n\) – since it must preserve the equation \(f(\alpha_i) = 0\).

4. The Galois Group¶

The Galois group of a field extension \(L/K\) is the group of all \(K\)-automorphisms of \(L\):

\[ \text{Gal}(L/K) = \{\sigma: L \to L \mid \sigma \text{ is an automorphism with } \sigma|_K = \text{id}\} \]

The operation is composition of maps.

Example: \(\mathbb{Q}(\sqrt{2})/\mathbb{Q}\)¶

The extension \(\mathbb{Q}(\sqrt{2}) = \{a + b\sqrt{2} \mid a, b \in \mathbb{Q}\}\) has degree \(2\). The only \(\mathbb{Q}\)-automorphisms are:

\(\text{id}: \sqrt{2} \mapsto \sqrt{2}\)
\(\sigma: \sqrt{2} \mapsto -\sqrt{2}\)

Hence \(\text{Gal}(\mathbb{Q}(\sqrt{2})/\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z}\).

Example: Splitting field of \(x^3 - 2\)¶

The roots of \(x^3 - 2\) are \(\sqrt[3]{2}\), \(\omega\sqrt[3]{2}\), and \(\omega^2\sqrt[3]{2}\) (with \(\omega = e^{2\pi i/3}\)). The splitting field is \(L = \mathbb{Q}(\sqrt[3]{2}, \omega)\) with \([L:\mathbb{Q}] = 6\).

The Galois group is \(\text{Gal}(L/\mathbb{Q}) \cong S_3\) – the symmetric group on \(3\) elements. This is because the automorphisms can arbitrarily permute the three roots (subject to the constraint that \(\omega \mapsto \omega\) or \(\omega \mapsto \omega^2\)).

Example: Cyclotomic fields¶

The \(p\)-th cyclotomic field \(\mathbb{Q}(\zeta_p)\) (with \(\zeta_p = e^{2\pi i/p}\)) has the Galois group:

\[ \text{Gal}(\mathbb{Q}(\zeta_p)/\mathbb{Q}) \cong (\mathbb{Z}/p\mathbb{Z})^* \]

This is the group of units modulo \(p\) – an abelian group of order \(p - 1\). Each automorphism \(\sigma_a\) sends \(\zeta_p \mapsto \zeta_p^a\) for some \(a \in \{1, \ldots, p-1\}\).

5. The Fundamental Theorem of Galois Theory¶

The fundamental theorem is the heart of the theory. It establishes a perfect correspondence between the algebraic structure of the field extension and the group structure of the Galois group.

Theorem (Galois correspondence). Let \(L/K\) be a finite Galois extension with Galois group \(G = \text{Gal}(L/K)\). Then there is an inclusion-reversing bijection:

\[ \{\text{intermediate fields } K \subseteq M \subseteq L\} \longleftrightarrow \{\text{subgroups } H \leq G\} \]

given by:

\[ M \longmapsto \text{Gal}(L/M), \qquad H \longmapsto L^H = \{x \in L \mid \sigma(x) = x \text{ for all } \sigma \in H\} \]

Moreover: - \([M : K] = [G : H]\) (index of the subgroup = degree of the extension) - \(M/K\) is a Galois extension if and only if \(H \trianglelefteq G\) (normal subgroup) - In that case, \(\text{Gal}(M/K) \cong G/H\)

The correspondence visualised

Fields                  Groups
L                       {e}
|                        |
M₂                     H₂
|    \                 /    |
M₁     M₃          H₁    H₃
|    /                 \    |
K                       G

Larger fields correspond to smaller subgroups (and vice versa).

6. Solvability¶

Galois' original question was: when can an equation be solved by radicals? The fundamental theorem provides the answer.

Definition. A group \(G\) is called solvable if it possesses a chain of subgroups:

\[ \{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G \]

where each factor \(G_{i+1}/G_i\) is abelian (cyclic of prime order).

Theorem (Galois). A polynomial \(f \in K[x]\) is solvable by radicals if and only if its Galois group is solvable.

Consequence for degree \(\geq 5\): The symmetric group \(S_n\) is not solvable for \(n \geq 5\) (because the alternating group \(A_n\) is simple and non-abelian for \(n \geq 5\)). Since there exist polynomials with Galois group \(S_5\), these are not solvable by radicals.

Consequence for degree \(\leq 4\): The groups \(S_1, S_2, S_3, S_4\) are all solvable – hence solution formulas exist for polynomials up to degree \(4\).

7. The Absolute Galois Group¶

For number theory – and in particular for Wiles' proof – it is not the Galois group of a single extension that matters, but the absolute Galois group:

\[ G_{\mathbb{Q}} = \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \]

This is the Galois group of the extension of all algebraic numbers over \(\mathbb{Q}\). It is an infinite, profinite group – the projective limit of all finite Galois groups \(\text{Gal}(L/\mathbb{Q})\).

\(G_{\mathbb{Q}}\) is one of the most mysterious objects in mathematics. Despite intensive research, its complete structure is unknown. What we do know are its representations – homomorphisms from \(G_{\mathbb{Q}}\) into matrix groups.

Galois Representations¶

A (continuous) Galois representation is a homomorphism:

\[ \rho: G_{\mathbb{Q}} \to \text{GL}_n(K) \]

for a suitable field or ring \(K\). For Wiles' proof, the two-dimensional representations (\(n = 2\)) are central:

\[ \rho: G_{\mathbb{Q}} \to \text{GL}_2(\mathbb{F}_p) \quad \text{or} \quad \rho: G_{\mathbb{Q}} \to \text{GL}_2(\mathbb{Z}_p) \]

Every elliptic curve \(E\) over \(\mathbb{Q}\) yields such representations – via the action of \(G_{\mathbb{Q}}\) on the \(p\)-division points \(E[p]\). And every modular form likewise yields a Galois representation. The Taniyama–Shimura conjecture states in essence: the representation of the elliptic curve is the representation of a modular form.

Local Galois Groups¶

For every prime \(p\) there is a local Galois group \(G_{\mathbb{Q}_p} = \text{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\). It controls the behaviour of algebraic objects "at the place \(p\)". Every global representation \(\rho: G_{\mathbb{Q}} \to \text{GL}_n(K)\) induces by restriction local representations:

\[ \rho|_{G_{\mathbb{Q}_p}}: G_{\mathbb{Q}_p} \to \text{GL}_n(K) \]

The local-global principle asks: when do the local representations determine the global one? This question lies at the centre of Wiles' proof.