Theory of Algebraically Closed Fields of Characteristic p is Complete

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $p$ be either $0$ or a prime number.

Let $ACF_p$ be the theory of algebraically closed fields of characteristic $p$ in the language $\mathcal L_r = \left\{ {0, 1, +, -, \cdot}\right\}$ for rings, where:

$0, 1$ are constants

and:

$+, -, \cdot$ are binary functions.


Then $ACF_p$ is complete.


Proof

By the Łoś-Vaught Test, it suffices to show that $ACF_p$ is satisfiable, has no finite models, and is $\kappa$-categorical for some uncountable $\kappa$.


Satisfiability

$\C$ is an algebraically closed field of characteristic $0$.

If $p$ is a prime, then the algebraic closure of $\Z / \Z_p$ is an algebraically closed field of characteristic $p$.

Thus $ACF_p$ is satisfiable.



$\Box$


No Finite Models

From Algebraically Closed Field is Infinite:

$ACF_p$ has no finite models

$\Box$


$\kappa$-Categorical

From:

Field of Uncountable Cardinality $\kappa$ has Transcendence Degree $\kappa$

and:

Algebraically Closed Fields are Isomorphic iff they have the same Characteristic and Transcendence Degree

it follows that:

$ACF_p$ is $\kappa$-categorical for all uncountable $\kappa$

Hence:

$ACF_p$ is $\kappa$-categorical for some uncountable $\kappa$

as was to be proved.

$\blacksquare$