MATH 612 Due Wednesday April 28, 1993

1. Let k be a field. Suppose p = char k ≠ 0.

Fact: There exists an extension field of k denoted k^{\frac{1}{p}} which consists precisely of the pth roots of elements of k.

2. Prove that if F is an algebraic extension of k which is not separable, then k^{\frac{1}{p}} ⊗_{k} F contains non-trivial nilpotent elements. In particular, k^{\frac{1}{p}} is not linearly disjoint from F.

a) Prove that if F is any extension of k then the nil radical of k^{\frac{1}{p}} ⊗_{k} F is the unique maximal ideal of k^{\frac{1}{p}} ⊗_{k} F: in fact, if x = ∑ e_{i} ⊗ f_{i} ∈ k^{\frac{1}{p}} ⊗_{k} F and x^{p} ≠ 0, then x is invertible in k^{\frac{1}{p}} ⊗_{k} F.

b) Prove that if F is a separable extension of k and E is any extension of k, then E ⊗_{k} F has trivial nil radical. (hint: Reduce to the case where [F : k] is finite and also use Theorem 6.1, p. 290.)

c) Prove that if F is a separable extension of k then k^{\frac{1}{p}} ⊗_{k} F is a field.