Algebraic interior of a field extension

Let be a field extension. The algebraic interior is the set of all elements of algebraic over ,¹ #m/def/field moreover this is a field and intermediate extension so that

is a tower of field extensions.

Proof

Suppose are algebraic over . Then is algebraic by ^P1, so in particular is algebraic.

Properties

Let be a field extension.

Proof of 1

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

and since Compositions only of algebraic extensions are algebraic, is an algebraic extension, and in particular is algebraic over . But then 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. ↩