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A $C_1$ field is a Ci field with the bound $N>d$. The Chevalley-Warning theorem gives finite-field examples; the Tsen theorem gives function fields of curves over algebraically closed fields. A $C_1$ field has trivial Brauer group.

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.