Algebraic extension

Compositions only of algebraic extensions are algebraic

Let be a tower of field extensions. Then is algebraic iff both and are algebraic.¹ field

Proof

If is algebraic, then every element of is algebraic over ; therefore and are algebraic.

Conversely, suppose and are algebraic, and let . Then there exists a polynomial

such that , whence is algebraic over the subfield

so

is finite. Now

is finite by ^P1 since all the are algebraic over by construction. Thus by the basic property of an Intermediate field extension,

is finite. To summarize, we have the tower

where squiggly lines are algebraic and dashed lines are finite. Finally we see that

must be finite and thus is algebraic over .

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Footnotes

1. 2009. Algebra: Chapter 0, §VII.1.3, p. 395 ↩