Galois Theory: The Connection Between Fields and Groups

Galois Theory

Main Idea

Évariste Galois in 1830 established a deep connection between field extensions and groups. To each extension $K/F$, one associates the Galois group $\mathrm{Gal}(K/F)$ — the group of automorphisms of the field $K$ that fix $F$.

Field Extensions

$F \subseteq K$ — a field extension. The degree $[K:F] = \dim_F K$.

$\mathbb{Q} \subseteq \mathbb{R} \subseteq \mathbb{C}$ — a chain of extensions. $[\mathbb{C}:\mathbb{R}] = 2$, $[\mathbb{R}:\mathbb{Q}] = \infty$.

$\mathbb{Q}(\sqrt{2}) = {a + b\sqrt{2}: a, b \in \mathbb{Q}}$ — an extension of degree $2$. The minimal polynomial of $\sqrt{2}$ over $\mathbb{Q}$: $x^2 - 2$.

Galois Group

$\mathrm{Gal}(K/F) = {\sigma: K \to K \mid \sigma \text{ is an automorphism},\ \sigma|_F = \mathrm{id}}$.

For $K = \mathbb{Q}(\sqrt{2})$: $\mathrm{Gal}(K/\mathbb{Q}) = {\mathrm{id},\ \sigma}$, where $\sigma(a + b\sqrt{2}) = a - b\sqrt{2}$. A group of order $2$, $\cong \mathbb{Z}_2$.

For $K = \mathbb{Q}(\zeta)$ ($\zeta = e^{2\pi i / p}$, $p$ is prime): $\mathrm{Gal}(K/\mathbb{Q}) \cong (\mathbb{Z}/p\mathbb{Z})^*$ — a cyclic group of order $p-1$.

Fundamental Theorem of Galois Theory

For a normal extension $K/F$ (the splitting is normal, as in $\mathbb{C}/\mathbb{R}$):

There is a one-to-one correspondence between intermediate fields $F \subseteq E \subseteq K$ and subgroups $H \leq \mathrm{Gal}(K/F)$: $E \leftrightarrow \mathrm{Gal}(K/E)$.

Remark: larger fields correspond to smaller subgroups and vice versa.

Solvability by Radicals

The field $K$ is obtained from $F$ by iterating extensions of the form $F(\sqrt[n]{a})$ (adjoining an $n$th root). This corresponds to normal subgroups with factors $\mathbb{Z}/m\mathbb{Z}$ — solvable series.

An equation is solvable by radicals $\iff$ its Galois group is solvable.

$S_5$ is not solvable $\rightarrow$ the general equation of degree $5$ is not solvable. This is the answer to a question that occupied mathematicians for $250$ years.