Proposition 1

— Let $\Omega$ be an extension of a field $K$ and let $L, M$ be two subextensions of $\Omega$ which are algebraically disjoint over $K$. Suppose that the relative separable algebraic closure of $K$ in $L$ is equal to $K$^1. Let $\varphi$ be the $K$-algebra homomorphism of $L \otimes_K M$ into $\Omega$ mapping $x \otimes y$ to $xy$ for $x \in L, y \in M$, and let $p$ be the kernel of $\varphi$. Then $p$ is the set of nilpotent elements of $L \otimes_K M$ and this is the smallest prime ideal of $L \otimes_K M$.

On replacing $\Omega$ by an algebraic closure if necessary, we may suppose $\Omega$ algebraically closed. Let $B$ be a transcendence basis of $M$ over $K$, $N$ the relative algebraic closure of $K(B)$ in $\Omega$ and $N_r$ (resp. $N_s$) the set of elements of $N$ which are separable (resp. p-radical) over $K(B)$. We remark that $M$ is algebraic over $K(B)$, so that $N$ is the relative algebraic closure of $M$ in $\Omega$.

Fig. 1.

Let us define the following chain of homomorphisms:

We have $\alpha = \mathrm{Id}_L \otimes u$ where $u$ is the canonical injection of $M$ into $N$, hence $\alpha$ is injective. The mapping $\beta$ is the isomorphism of commutative groups which maps $x \otimes y$ to $(x \otimes 1) \otimes y$ (II, p. 278, Prop. 2) for $x \in L$ and $y \in N$. We have $\gamma = v \otimes \mathrm{Id}_N$, where $v$ is the $K$-algebra homomorphism of $L \otimes_K K(B)$ into $L(B)$ which maps $x \otimes y$ to $xy$, for $x \in L$ and $y \in K(B)$; since $L$ and $M$ are algebraically disjoint over $K$, Prop. 14 (V, p. 114) shows that $L$ and $K(B)$ are linearly disjoint over $K$, in other words, $v$ is injective, hence $y$ is injective. Since $N$ is a quasi-Galois extension of $K(B)$, there exists (V, p. 76, Prop. 13) a $K(B)$-algebra isomorphism $w$ of $N_s \otimes_{K(B)} N_r$ onto $N$ mapping $x \otimes y$ to

1 This hypothesis is sometimes expressed by saying that $L$ is a primary extension of $K$.

xy for $x \in \mathbf{N}$, and $y \in \mathbf{N}_r$; we denote by 6 the isomorphism $\mathrm{Id}_{L(B)} \otimes w \ ^1$. By Th. 1 (V, p. 137) and the hypothesis on the extension L of K, every element of $L(B)$ which is algebraic and separable over $K(B)$ belongs to $K(B)$; in particular, we have $L(B) \cap N_s = K(B)$. Since $N_r$ is a Galois extension of $K(B)$, Th. 5 (V, p. 71) shows that there exists a $K(B)$-algebra isomorphism $w'$ of $L(B) \otimes_{K(B)} N_s$ onto $L(B U N_r)$ mapping $x \otimes y$ to $xy$ for $x \in L(B)$ and $y \in N_r$; we denote by $\varepsilon$ the isomorphism $w' \otimes \mathrm{Id}_{N_r}$. Finally, $\zeta$ is the K-algebra homomorphism mapping $x \otimes y$ to $xy$ for $x \in L(B U N_r)$ and $y \in N_r$.

What has been said shows that $\eta = \varepsilon \delta \gamma \beta \alpha$ is an injective K-algebra homomorphism of $L \otimes_K M$ into $L(B U N_r) \otimes_{K(B), N_r}$. Further, every element of M is of the form $\sum_{i=1}^n a_i b_i$ with $a_i \in \mathbf{N}$, and $b_i \in \mathbf{N}_r$ for $1 \leq i \leq n$; hence we obtain $\varphi = \zeta \eta$.

The kernel $p$ of $\varphi$ is a prime ideal of $L \otimes_K M$, hence every nilpotent element of $L \otimes_K M$ belongs to $p$, by Prop. 2 (V, p. 118). Conversely let $a$ be an element of $p$; put $\eta(a) = \sum_{r=1}^s b_i \otimes c_i$ with $b_i \in L(B U N_r)$ and $c_i \in N_r$ for $1 \leq i \leq s$. Since $N_r$ is a p-radical extension of $K(B)$, there exists an integer $f \geq 0$ such that $c_i^{p^f}$ belongs to $K(B)$ for $1 \leq i \leq s$ (where $p$ is the characteristic exponent of K). But we have

and since $\eta$ is injective we finally have $a^{p^f} = 0$. We have thus shown $p$ to be the set of all nilpotent elements of $L \otimes_K M$. Now every prime ideal of $L \otimes_K M$ contains $p$ by Prop. 2 (V, p. 118).