Symmetries that shuffle roots while respecting arithmetic -- the heart of the theory
The Galois group of a field extension E/F consists of all automorphisms of E that fix F pointwise. These are symmetries that shuffle roots of polynomials while respecting all arithmetic relations. The structure of this group -- whether it is abelian, solvable, or simple -- controls which equations can be solved by radicals.
In this lesson, watch automorphisms permute roots, build Cayley multiplication tables cell by cell, and see how the Galois group embeds inside the symmetric group S_n.
Roots of a polynomial sit in the complex plane. Each automorphism of the splitting field permutes these roots -- but not arbitrarily. Automorphisms must send each root to a conjugate root (one that satisfies the same minimal polynomial). Click the buttons below to watch roots animate to their new positions.
Galois group: Klein 4-group (V4) -- click an automorphism to animate
Key insight: An automorphism is completely determined by where it sends the generators of the extension. For Q(sqrt(2), sqrt(3))/Q, every automorphism is determined by the choice of sign for sqrt(2) and sqrt(3) independently, giving 2 x 2 = 4 automorphisms -- the Klein 4-group.
The Cayley table encodes the complete multiplication structure of a group. Click cells to compose two automorphisms: first apply the column element, then the row element. Color patterns in the completed table reveal subgroups, generators, and whether the group is abelian (symmetric table) or not.
Gal(Q(sqrt(2),sqrt(3))/Q) -- every element has order 2
| * | e | sigma | tau | sigma*tau |
|---|---|---|---|---|
| e | ? | ? | ? | ? |
| sigma | ? | ? | ? | ? |
| tau | ? | ? | ? | ? |
| sigma*tau | ? | ? | ? | ? |
Key insight: Compare V4 (abelian -- table is symmetric) with S3 (non-abelian -- table is asymmetric). The S3 table shows that rs != sr, which is exactly why the Galois group of x^3 - 2 is "harder" than that of x^2 - 2 or x^4 - 1. Non-commutativity is what ultimately connects to insolvability by radicals.
Every Galois group acts on the roots as permutations, giving an embedding Gal(E/F) <= S_n where n is the number of roots. This embedding may or may not be surjective: the Galois group could be the full symmetric group or a proper subgroup.
Gal is a proper subgroup of S_4 -- not every permutation is an automorphism
Key insight: Algebraic relations between roots constrain which permutations are valid automorphisms. For x^4 - 5x^2 + 6 = (x^2-2)(x^2-3), the Galois group is V4 inside S4 (only 4 of the 24 permutations), because an automorphism sending sqrt(2) to sqrt(3) would violate the relation (sqrt(2))^2 = 2 != 3 = (sqrt(3))^2.