MagicInternetMath on Nostr: 📐 Galois Let $f(x) = 0$ be an equation with distinct roots whose Galois group over ...

📐 Galois

Let $f(x) = 0$ be an equation with distinct roots whose Galois group over $K$ is $G$. Then $f(x) = 0$ can be solved by radicals if and only if $G$ is solvable -- that is, has a composition series $G \\supset G_1 \\supset G_2 \\supset \\cdots \\supset G_\\nu = \\{e\\}$ in which each $G_i$ is a normal subgroup of prime index in its predecessor.

Proof: Necessity: If solvable by radicals, the tower of field extensions reduces the Galois group at each step to a normal subgroup of prime index (by the proposition of \u00a744). Taking only steps where the group decreases gives the composition series. Sufficiency: If $G$ is solvable, the proposition ...

From: gal-edwards
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