Algebraic element

Algebraic interior of a field extension

Let be a field extension. The algebraic interior is the set of all elements of algebraic over ,¹ 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.

1. If is algebraically closed, then is an algebraic closure of .

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.

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Footnotes

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