Field extension

Field extension of finite type

A field extension is finitely generated iff is the adjunction of finitely many elements to , 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

• Hilbert’s Nullstellensatz

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