Exercise 4, § 2

Let $K$ be a field, $\Omega$ an extension of $K$ and $E, F$ linearly disjoint subextensions over $K$. Let $C$ be the $K$-algebra $K[E \cup F]$ isomorphic to $E \otimes_K F$. a) Let $E'$ (resp. $F'$) be a subextension of $E$ (resp. $F$) and put $C' = K[E' \cup F']$. If an element $a$ of $C'$ is invertible in $C$ then $a^{-1} \in C'$. (Reduce to the case $F' = F$. Let

E" be a supplement of E' in the vector E'-space E, so that C is isomorphic to $C' \oplus (E'' \otimes_K F)$. Express the matrix of multiplication by a in this decomposition.) \* b) Show that if C is a field, then E or F is algebraic over K. (If $x \in E$ and $y \in F$ are transcendental over K, take $E' = K(x)$, $F' = K(y)$ and $a = x + y$ in $a$.)