Remark 2

Let L be a commutative field with the following property: $(C_1)$ Let V be a vector space of finite dimension $n$ over the field L, and let $d$ be such that $0< d < n$. For every homogeneous polynomial mapping $F : V\rightarrow L$ of degree $d$, there exists a nonzero element $x$ of V such that $F(x) = 0$.

The proof of Theorem 2 shows that every field of finite degree over L with center L is equal to L.

By the corollary of VIII, p. 356, every finite field has property $(C_1)$. We can prove (VIII, p. 360, Exercise 7) that the following fields have property $(C_1):$

– every algebraic extension of a finite field

– every field of rational fractions in one variable with coefficients in an algebraically closed field (Tsen’s theorem).

– $*$every field endowed with a discrete valuation for which it is complete and

whose residue field is algebraically closed (VIII, p. 332, Exercise 17)$.*$