Algebraically Closed Field

Let be the Language of fields. ACF is the Theory consisting of the Field axioms, as well as, for every , the sentence:

Additionally, let be the sentence

and set

and for every prime , set

Theorem

Both and (for prime ) are -Categorical for all .

Proof

Follows from the properties of Transcendence Degree.

Corollary

Both and (for prime ) are a Complete Theory.

Proof

Vaught’s Test.