CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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The Galois group of a cyclotomic extension is abelian

Statement

Let $K$ be a field and $n \geq 1$ with $char K ∤ n$ (The characteristic of a ring: the least $n \geq 1$ with $n \cdot 1_{R} = 0$ when one exists, and $0$ otherwise). Then $Gal (K (μ_{n}) / K)$ (The cyclotomic extension $K (μ_{n})$ as a splitting field of $t^{n} - 1$ ) is abelian.

Facts & Assumptions

Given: A field $K$ and $n \geq 1$ with $char K ∤ n$ .

[L1]

$K (μ_{n}) / K$ is finite Galois and $σ \mapsto [a_{σ}]_{n}$ is an injective group homomorphism $Gal (K (μ_{n}) / K) \to (Z / n)^{\times}$ ( $K (μ_{n}) / K$ is Galois and $σ \mapsto a_{σ}$ embeds its Galois group into $(Z / n)^{\times}$ ).

[L2]

For every $n \geq 1$ , $(Z / n, \cdot, [1]_{n})$ is a commutative monoid (For every natural $n$ , $(Z / n, +)$ is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold), and $(Z / n)^{\times}$ is the group of its invertible elements (The unit group $(Z / n)^{\times}$ and Euler's totient $φ (n) = ∣ (Z / n)^{\times} ∣$ for $n \geq 1$ ).

Proof

technique · direct

1.1

Multiplication on $Z / n$ is commutative by [L2], so the group $(Z / n)^{\times}$ , whose operation is that multiplication restricted to the units, is abelian.

2.1

By [L1] the group $Gal (K (μ_{n}) / K)$ is isomorphic to its image in $(Z / n)^{\times}$ , a subgroup of an abelian group; a subgroup of an abelian group is abelian, and a group isomorphic to an abelian group is abelian, so $Gal (K (μ_{n}) / K)$ is abelian.

step 1.1L1∎

Depends on

Used by

Dependency tree · two levels

39 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

K. Conrad, Cyclotomic Extensions (expository blurb), Section 2 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, v5.10, Proposition 5.8 (standard reference, not scraped)