Exercise 7, § 18

a) Prove that if the field K has property $(C_1)$ (VIII, p. 357), then so does every algebraic extension of K. (Reduce to the case of an extension L of K of finite degree $r$. Prove that if F is a homogeneous polynomial function of degree $d$ on an L-vector space V, then the function $N_{L/K}\circ F$ is homogeneous polynomial of degree $rd$ on the K-vector space V.)

$*$b) Assume from now on that K is algebraically closed. Let $F_1, . . . ,F_n$ be homogeneous polynomials of degree $>0$ in $m$ variables. Prove that if $n < m$, then there exists an $\mathbf{a}\not= 0$ in $K^n$ such that $F_1(\mathbf{a}) =\cdots = F_n(\mathbf{a}) = 0$ (use AC, VIII, §2, n$^o4$, p. 19, corollaire 1 du théorème 3 and AC, VIII, §3, n$^o1$, p. 24, proposition 2).

c) Deduce from b) that the field K(X) has property $(C_1)$.

(Let $f$ be a homogeneous polynomial of degree $d$ in $n$ variables whose coefficients are polynomial in X, and let $\nu$ be the maximum of the degrees of these coefficients. Let N be an integer, and let $G_1, . . . ,G_n$ be polynomials in X of degree $\leqslant N$. Prove that the equality $F(G_1(X), . . . ,G_n(X)) = 0$ is equivalent to the annihilation of $Nd+\nu + 1$ homogeneous polynomials of degree $>0$ in the $n(N + 1)$ coefficients of the $G_i.$)

d) Deduce from a) and c) that an extension of an algebraically closed field of transcendence degree $\leqslant 1$ has property $(C_1)$ (“Tsen’s theorem”).$*$