Exercise 15, § 12

a) Let K be a commutative field, A a K-algebra, and $B = K[X_{\iota}]_{\iota\in I}$ a polynomial algebra over K. Prove that if every nonzero element of A is regular, then the algebra $A\otimes_KB$ has the same property.

b) Deduce that if D is a field of finite rank over K and E is a purely transcendental extension of K, then the ring $D\otimes_KE$ is a field.