Symmetries that shuffle roots while respecting arithmetic -- the heart of the theory
The roots of a polynomial can be shuffled in ways that no equation with rational coefficients can detect. Those shuffles form a group, and its shape decides whether the equation can be solved.
Choose an extension, then click an automorphism to watch the roots move to their new positions. Notice each root only ever lands on a conjugate root.
Galois group: Klein 4-group (V4) -- click an automorphism to animate
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.
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.
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.
Pick a group and click cells to compose two automorphisms (column first, then row), or press Reveal All. Compare V4, whose table is symmetric, with S3, whose table is not.
Gal(Q(sqrt(2),sqrt(3))/Q) -- every element has order 2
| * | e | sigma | tau | sigma*tau |
|---|---|---|---|---|
| e | ? | ? | ? | ? |
| sigma | ? | ? | ? | ? |
| tau | ? | ? | ? | ? |
| sigma*tau | ? | ? | ? | ? |
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.
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.
Group structure controls solvability -- abelian groups give solvable-by-radicals; non-abelian simple groups do not
Choose an example and click each automorphism to see it as a permutation of the roots. Notice how few of the possible permutations are allowed.
Gal is a proper subgroup of S_4 -- not every permutation is an automorphism
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.
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.
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