The Galois Group
The Galois group of a polynomial is the group of permutations of its roots that preserve all algebraic relations among them. It is the bridge between algebra (field extensions) and symmetry (group theory) — and the key to understanding which equations can be solved by radicals.
The quadratic case
Consider $x^2 - 2 = 0$, with roots $x_1 = \sqrt{2}$ and $x_2 = -\sqrt{2}$.
Over $\mathbb{Q}$, we can’t tell the roots apart — all we know is $x_1 + x_2 = 0$ and $x_1 x_2 = -2$, both of which are invariant under swapping. The Galois group is ${e, (12)} \cong \mathbb{Z}_2$.
When we extend to $\mathbb{Q}(\sqrt{2})$, we can express $x_1 = \sqrt{2}$ explicitly. Now the roots are distinguishable, and the Galois group collapses to ${e}$.
Pattern: extending the field → distinguishing roots → shrinking the Galois group.
The quartic correspondence
This is where the magic becomes visible. Consider the quartic
$$x^4 + px^2 + q = 0$$
with four roots $x_1, x_2, x_3, x_4$. Following the approach in Varriest’s example, we can track exactly how field extensions and symmetry groups move in opposite directions:
At each step, we adjoin a new element (grow the field) and identify which permutations still preserve all known relations (shrink the group):
| Field | New element | Relations | Symmetry group | Order |
|---|---|---|---|---|
| $\mathbb{Q}(p, q)$ | — | $x_1+x_2=0$, $x_3+x_4=0$ | $D_4$ | 8 |
| $\mathbb{Q}(p,q,w_1)$ | $w_1 = \sqrt{p^2-4q}$ | $x_1^2 - x_3^2 = w_1$ | $K = {e,(12),(34),(12)(34)}$ | 4 |
| $\mathbb{Q}(p,q,w_1,w_2)$ | $w_2 = \sqrt{\frac{-p-w_1}{2}}$ | $x_3 - x_4 = 2w_2$ | $\mathbb{Z}_2 = {e, (12)}$ | 2 |
| $\mathbb{Q}(p,q,w_1,w_2,w_3)$ | $w_3 = \sqrt{\frac{-p+w_1}{2}}$ | $x_1 - x_2 = 2w_3$ | $E = {e}$ | 1 |
The field tower grows upward: $\mathbb{Q} \subset F_1 \subset F_2 \subset F_3 \subset F_4$.
The group chain shrinks downward: $S_4 \supset D_4 \supset K \supset \mathbb{Z}_2 \supset E$.
This is the Galois correspondence: there’s an order-reversing bijection between intermediate fields and subgroups of the Galois group. Bigger field ↔ smaller group.
Galois’s insight
Evariste Galois (1811–1832) discovered this correspondence at age 19. He showed that whether a polynomial is solvable by radicals — expressible using only $+, -, \times, \div$ and roots — depends entirely on the structure of its Galois group.
Each radical extension (adjoining $\sqrt[n]{a}$) corresponds to a step in the group chain. But not just any step — the subgroups must be normal, and the quotients must be cyclic. When these conditions hold, you can peel off the roots layer by layer. When they don’t, you’re stuck.
The question “can we solve this equation?” becomes the question “does its Galois group have the right kind of subgroup chain?” This is the concept of a solvable group.
Why it matters
The Galois group is one of the most powerful ideas in mathematics: - It proves the impossibility of certain constructions (no quintic formula, no trisecting angles with straightedge and compass) - Galois representations (how Galois groups act on vector spaces) are the bridge between elliptic curves and modular forms — the key to the modularity theorem and Fermat’s Last Theorem - It launched abstract algebra as a field — groups, rings, fields, all descended from Galois’s ideas
Galois wrote his key ideas in a letter the night before a duel in which he was killed, age 20. The mathematics he created in those pages took a century to fully develop.