C2 field ($C_2$)

A $C_2$ field is a Ci field with the bound $N>d^2$. Function fields of curves over finite fields are examples. The extra-variable construction in surjectivity of reduced norms over C2 fields makes this polynomial-solvability property useful for division algebras.

For a central division algebra $D$ of degree $d$ over a C2 field, the form $\mathrm{Nrd}(X)-a{T}^d$ has degree $d$ in $d^2+1$ variables. Its nontrivial zero must have $T\ne 0$, because the reduced norm of a division algebra is anisotropic. Dividing $X$ by $T$ gives norm $a$. Zero is the norm of zero.