Is the following statement true? Every extension of degree n has a Galois group of order n

No. ℚ(∛2)/ℚ has degree 3 but only the identity automorphism The extension is not normal — the other cube roots are not real. Order = degree only for Galois extensions.

Counterexample

Is this true?

Every extension of degree n has a Galois group of order n

No — it is false.

The counterexample

$\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ has degree 3 but only the identity automorphism

Why it fails

The extension is not normal — the other cube roots are not real. Order = degree only for Galois extensions.

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Base field or ring

The answer changes with the field or ring you are working over.

Irreducible over ℤ ⇒ irreducible mod every prime
An algebraic extension is a finite extension

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fields galois

Go deeper· Explained in full in Galois correspondence and the standard groups· Asked in CSIR NET December 2025· From Rings and Fields › Galois theory essentials

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.

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