finitely generated extension

Definition

A field extension is finitely generated if there exists such that

for progressive simple extensions.

We denote

Relation with algebraic

Let be finitely generated. Then the following are equivalent:

1. is finite.

2. is algebraic.

3. are algebraic over .

Proof

is true even if it’s not finitely generated.

is trivial (by definition)

: Let .

This can be proven inductively. First, we must show that

then we can use the transitivity of degree argument inductively to show

This can be proven more generally: Claim: Let and with . If is finite then

Proof of claim: Let be finite. Then by properties of simple field extensions,

for the minimal monic polynomial . This implies . since all coefficients are elements of , so when considering it over .

Intuition for proof

Since there are some linear relationship using coefficients in , then the same relationship will exist inside since it includes all the same coefficients. may not be minimal anymore, since there are more coefficients to work with, so there may be more linear relationships. This would only decrease the possibility of having linear independent basis vectors.