Algebraic interior of a field extension

Let $𝐿 : 𝐾$ be a field extension. The algebraic interior $(𝐿 : 𝐾) \circ$ is the set of all elements of $𝐿$ algebraic over $𝐾$ ,¹ #m/def/field moreover this is a field and intermediate extension so that

𝐿 | (𝐿 : 𝐾) \circ | 𝐾

is a tower of field extensions.

Proof

Suppose $𝛼, 𝛽 \in 𝐿$ are algebraic over $𝐾$ . Then $𝐾 (𝛼, 𝛽)$ is algebraic by ^P1, so in particular $𝛼 𝛽 - 1$ is algebraic.

Properties

Let $𝐿 : 𝐾$ be a field extension.

If $𝐿$ is algebraically closed, then $―― 𝐾 = (𝐿 : 𝐾) \circ$ is an algebraic closure of $𝐾$ .

Proof of 1

The extension $―― 𝐾 : 𝐾$ is tautologically algebraic, so we need only show that $―― 𝐾$ is algebraically closed. To this end let $𝛼$ be algebraic over $―― 𝐾$ , so

𝐾 : ―― 𝐾 : ―― 𝐾 (𝛼)

and since Compositions only of algebraic extensions are algebraic, $𝐾 : 𝐾 (𝛼)$ is an algebraic extension, and in particular $𝛼$ is algebraic over $𝐾$ . But then $𝛼 \in ―― 𝐾$ by definition of the latter.

#state/tidy | #lang/en | #SemBr

This is nonstandard terminology which I have not seen used elsewhere, but I like the analogy to Algebraic closure. ↩

Algebraic interior of a field extension

Properties

Footnotes