Math 129: Algebraic Number Theory Homework Assignment 10

William Stein

Due: Thursday, April 29, 2004

1. Prove that the ring C defined in Section 9 really is the tensor product of A and B, i.e., that it satisfies the defining universal mapping property for tensor products. Part of this problem is for you to look up a functorial definition of tensor product.

2. Find a zero divisor pair in Q(\sqrt{5}) ⊗_{Q} Q(\sqrt{5}).

3. (a) Is Q(\sqrt{5}) ⊗_{Q} Q(\sqrt{−5}) a field?

(b) Is Q(\sqrt[4]{5}) ⊗_{Q} Q(\sqrt[4]{−5}) ⊗_{Q} Q(\sqrt{−1}) a field?

4. Suppose ζ_{5} denotes a primitive 5th root of unity. For any prime p, consider the tensor product Q_{p} ⊗_{Q} Q(ζ_{5}) = K_{1} ⊕ ⋯ ⊕ K_{n(p)}. Find a simple formula for the number n(p) of fields appearing in the decomposition of the tensor product Q_{p} ⊗_{Q} Q(ζ_{5}). To get full credit on this problem your formula must be correct, but you do not have to prove that it is correct.

5. Suppose ∥ · ∥_{1} and ∥ · ∥_{2} are equivalent norms on a finite-dimensional vector space V over a field K (with valuation | · |). Carefully prove that the topology induced by ∥ · ∥_{1} is the same as that induced by ∥ · ∥_{2}.

6. Suppose K and L are number fields (i.e., finite extensions of Q). Is it possible for the tensor product K ⊗_{Q}L to contain a nilpotent element? (A nonzero element a in a ring R is nilpotent if there exists n > 1 such that a^{n} = 0.)

7. Let K be the number field Q(\sqrt[5]{2}).

(a) In how many ways does the 2-adic valuation | · |_{2} on Q extend to a valuation on K?

(b) Let v = | · | be a valuation on K that extends | · |_{2}. Let K_{v} be the completion of K with respect to v. What is the residue class field F of K_{v}?