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  "body": "\n## Definition\n\nLet $K$ be a [[field|0QA8CHH8MG]].\nThe field $K$ is *quasi-algebraically closed*, or a *$C_1$ field*, if the following holds.\nFor all integers $n \u003e d \\ge 1$, every [[homogeneous polynomial|SD3M01318E]] $f \\in K[x_1, \\ldots, x_n]$ of degree $d$ has a zero $(a_1, \\ldots, a_n) \\in K^n$ with $(a_1, \\ldots, a_n) \\ne (0, \\ldots, 0)$.\n\n## Remarks\n\nEvery [[algebraically closed field|02SJG1HC32]] is $C_1$.\nIndeed, let $g(t) = f(t, 1, 0, \\ldots, 0)$.\nIf $g$ is not constant, it has a root $t_0$, and $(t_0, 1, 0, \\ldots, 0)$ is a nontrivial zero of $f$.\nIf $g$ is constant, the coefficient of $x_1^d$ in $f$ is $0$, so $(1, 0, \\ldots, 0)$ is a nontrivial zero of $f$.\n\nExample: a [[finite field|08RTDHX1SK]] $\\F_q$ is $C_1$.\nCheck: by the [[Chevalley–Warning theorem|08895Q5W96]], the number of zeros in $\\F_q^n$ of a polynomial of degree $d \u003c n$ is divisible by the characteristic.\nThe zero vector is one zero, so there is another.\n\nNon-example: $\\R$ is not $C_1$.\nThe form $x^2 + y^2 + z^2$ has degree $2$ in $3$ variables, but its only real zero is $(0, 0, 0)$.\n\nMore generally, $K$ is a $C_i$ field if every homogeneous polynomial of degree $d \\ge 1$ in $n \u003e d^i$ variables has a nontrivial zero; $C_1$ is the case $i = 1$.\n",
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