---
tag: 92SQ10PN6J
kind: concept
name: Pseudo algebraically closed field
msc: ["12E30", 12L12, 14G05]
sources:
  - url: https://en.wikipedia.org/wiki/Pseudo_algebraically_closed_field
same_as:
  wikidata: Q7254461
  wikipedia: Pseudo algebraically closed field
---

## Summary

A field $K$ is pseudo algebraically closed, or PAC, if every absolutely irreducible algebraic variety defined over $K$ has a point with coordinates in $K$.
[[Algebraically closed fields|0PF6TM8TWT]] are PAC, but many PAC fields are far from algebraically closed.
PAC fields are a central notion of field arithmetic, and their theory links Galois theory with model theory through the [[absolute Galois group|PGKWBJDTKX]].

## Explanations

### L1 human

An algebraically closed field contains a root of every polynomial equation in one unknown.
A PAC field asks for something different.
Every system of polynomial equations in several unknowns whose solution set is geometrically "one piece" must have a solution in the field.
The field of rational numbers is not PAC, since $x^2 + y^2 = -1$ has no rational solution although its solution set over $\C$ is one irreducible curve.
Examples of PAC fields include the infinite algebraic extensions of a finite field.