Is there are nice way to prove the primitive element theorem without using field extensions? The primitive element theorem says that if $x$ and $y$ are algebraic over $F$ and $y$ is separable over ...
Several questions actually. All rings and algebras are supposed to be commutative and with $1$ here. (1) Let $k$ be a field, and let $A$ and $B$ be two $k$-algebras. I need a proof that if $A$ and ...