Exercise 4, § 7

Let K be a field, E = K(x) a separable algebraic extension of K and f the minimal polynomial of x in K[X].

a) Let F be an extension of K; in the ring F[X] the polynomial f decomposes into a product f_1 f_2 \ldots f_r of irreducible and separable pairwise distinct polynomials. Show that the algebra E \otimes_K F is isomorphic to the product of fields F[X]/(f_j) (1 \leq j \leq r).

b) Suppose that F = K(y) is a separable algebraic extension of K; let g be the minimal polynomial of y in K[X], and let g = g_1 g_2 \ldots g_s be the decomposition of g into a product of irreducible polynomials in $E[X]$. Show that $r = s$ and that if $m = \deg f,\ n = \deg g,\ m_j = \deg f_j,\ n_j = \deg g_j$ we can, after permuting the $g_j$ if necessary, suppose that $m_j / n_j = m / n$ for $1 \leq j \leq r$.