---
tag: G7RM16N4PF
kind: concept
name: Quasi-algebraically closed field
msc: ["12E30", 11D72]
sources:
  - url: https://en.wikipedia.org/wiki/Quasi-algebraically_closed_field
same_as:
  wikidata: Q283684
  wikipedia: Quasi-algebraically closed field
---

## Summary

A field is quasi-algebraically closed, or $C_1$, if every [[homogeneous polynomial|2C3PAQ7H50]] with more variables than its degree has a zero other than the zero vector.
Every [[algebraically closed field|0PF6TM8TWT]] is $C_1$, and so is every [[finite field|0ZXZ82FTX2]], by the [[Chevalley–Warning theorem|08895Q5W96]].
By [[Tsen's theorem|V8HDYN3JMY]], the function field of a curve over an algebraically closed field is $C_1$.
The [[Brauer group|BEW48TA9WZ]] of a $C_1$ field is trivial, which is one reason the notion matters.

## Explanations

### L1 human

Over an algebraically closed field, a polynomial in one variable always has a root.
A $C_1$ field has a weaker property for polynomials in many variables: if a form has many variables compared with its degree, it has a nonzero solution.
For instance, over a finite field every quadratic form in $3$ variables has a nonzero zero.
Over $\R$ this fails: $x^2 + y^2 + z^2$ is zero only at the origin.
So $\R$ is not $C_1$, while the finite fields are.