Subfield

A subfield $𝐹$ of a field $𝐾$ is a subring which is itself a field. field Often we consider these in terms of field extensions.

Tests for subfields

Theorem. Let $𝐹 \subseteq 𝐾$ . Iff for all $𝑎, 𝑏 \in 𝐹$ , we have $𝑎 - 𝑏 \in 𝐹$ and $𝑏 \neq 0$ implies $𝑎 𝑏 - 1 \in 𝐾$ , then $𝐹$ is a subfield. field

Proof

By Subrng test, we have a subrng. Since $1 \cdot (1) - 1 = 1 \in 𝐹$ , we have a subring. Commutativity of $𝐾$ implies commutativity of $𝐹$ , and every nonzero element has a unit. Therefore $𝐹$ is a field.

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Subfield

Tests for subfields

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