Field extension

Field extension of finite type

A field extension is finitely generated iff is the adjunction of finitely many elements to , #m/def/field i.e.

for some . Equivalently, is the composite of finitely many intermediate simple extensions.

Properties

Suppose is of finite generated, so

  1. is a finite field extension iff it is an algebraic extension iff are algebraic.
Proof of 1

If is finite it then it is automatically algebraic and the generators are also.

Suppose then that is algebraic, and thus each of the are algebraic, say of degree . Since intermediate extensions are all finite of degree , it follows so is , and thus is finite.

Other results

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