CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Every polynomial of degree at most four is solvable by radicals

Statement

Every polynomial over a field of characteristic $0$ of degree at most four is solvable by radicals.

Facts & Assumptions

Given: A characteristic- $0$ field $F$ and a polynomial $f \in F [x]$ of degree at most four.

[L1]

The groups $S_{n}$ are solvable for $n \leq 4$ (The symmetric groups $S_{n}$ are solvable for $n \leq 4$ ).

[L2]

The Galois group of a separable degree- $m$ polynomial embeds in $S_{m}$ (A polynomial Galois group acts faithfully on its roots).

[L3]

Subgroups of solvable groups are solvable (Subgroups and quotients of solvable groups are solvable).

[L4]

In characteristic $0$ , a polynomial with solvable Galois group is solvable by radicals (In characteristic $0$ , a solvable Galois group makes a polynomial solvable by radicals).

Proof

technique · direct

1.1

Let $E / F$ be the splitting field of $f$ , and let $G$ be its Galois group. Since $f$ is separable in characteristic $0$ , [L2] embeds $G$ in $S_{m}$ for some $m \leq 4$ . By [L1] and [L3], the group $G$ is solvable.

L1L2L3

2.1

Apply [L4] to $G$ : the polynomial $f$ is solvable by radicals.

step 1.1L4∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, Fields and Galois Theory, v5.10, Section 7 (standard reference, not scraped)
J. Ash, Basic Abstract Algebra, Section 6.8 (standard reference, not scraped)