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  "body": "\n## Summary\n\nA 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.\nEvery [[algebraically closed field|0PF6TM8TWT]] is $C_1$, and so is every [[finite field|0ZXZ82FTX2]], by the [[Chevalley–Warning theorem|08895Q5W96]].\nBy [[Tsen's theorem|V8HDYN3JMY]], the function field of a curve over an algebraically closed field is $C_1$.\nThe [[Brauer group|BEW48TA9WZ]] of a $C_1$ field is trivial, which is one reason the notion matters.\n\n## Explanations\n\n### L1 human\n\nOver an algebraically closed field, a polynomial in one variable always has a root.\nA $C_1$ 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.\nFor instance, over a finite field every quadratic form in $3$ variables has a nonzero zero.\nOver $\\R$ this fails: $x^2 + y^2 + z^2$ is zero only at the origin.\nSo $\\R$ is not $C_1$, while the finite fields are.\n",
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