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Counterexample

Is this true?

An algebraic extension is a finite extension

No — it is false.

The counterexample

The field of all algebraic numbers Q̄ over Q\mathbb{Q}

Why it fails

Every element is algebraic, but the extension has infinite degree (it contains Q(21/n)\mathbb{Q}(2^{1/n}) for every n).

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Base field or ring

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

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Go deeper· Explained in full in Field extensions, degrees and splitting fields· Asked in CSIR NET June 2025· From Rings and Fields › Field extensions, splitting fields, finite fields

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