Field Generated by Finitely Many Algebraic Elements is Algebraic

Theorem

Let F be a field and K=F(α1,,αn) be a field extension where α1,,αn are algebraic over F. Then K/F as finite.

Proof

First note that F(α1)/F is finite due to the power basis.

Then, if F(α1,,αk) is finite so is F(α1,,αk)(αk+1)=F(α1,,αk+1) since αk+1 is algebraic over F which is a subfield of F(α1,,αk).

Thus the result follows by induction on k.