Math 129: Algebraic Number Theory Homework Assignment 9

William Stein

Due: Thursday, April 22, 2004

There are SIX problems.

1. In this problem you will compute an example of weak approximation, like I did in the example in Section 6 of lecture 18. Let K = Q, let | · |_{7} be the 7-adic absolute value, let | · |_{11} be the 11-adic absolute value, and let | · |_{∞} be the usual archimedean absolute value. Find an element b ∈ Q such that |b − a_{i}|_{i} < \frac{1}{10}, where a_{7} = 1, a_{11} = 2, and a_{∞} = −2004.

2. Prove that −9 has a cube root in Q_{10} using the following strategy (this is a special case of Hensel’s Lemma, which you can read about in an appendix to Cassel’s article).

(a) Show that there is an element α ∈ Z such that α^{3} ≡ 9 (mod 10^{3}).

(b) Suppose n ≥ 3. Use induction to show that if α_{1} ∈ Z and α^{3} ≡ 9 (mod 10^{n}), then there exists α_{2} ∈ Z such that α_{2}^{3} ≡ 9 (mod 10^{n+1}). (Hint: Show that there is an integer b such that (α_{1} + b · 10^{n})^{3} ≡ 9 (mod 10^{n+1}).)

(c) Conclude that 9 has a cube root in Q_{10}.

3. Prove that every finite extension of Q_{p} “comes from” an extension of Q, in the following sense. Given an irreducible polynomial f ∈ Q_{p}[x] there exists an irreducible polynomial g ∈ Q[x] such that the fields

Q_{p}[x]/(f)

and Q_{p}[x]/(g) are isomorphic. [Hint: Choose each coefficient of g to be sufficiently close to the corresponding coefficient of f, then use Hensel’s lemma to show that g has a root in Q_{p}[x]/(f).] 4. Find the 3-adic expansion to precision 4 of each root of the following

polynomial over Q_{3}:

f = x^{3} − 3x^{2} + 2x + 3 ∈ Q_{3}[x].

Your solution should conclude with three expressions of the form

a_{0} + a_{1} · 3 + a_{2} · 3^{2} + a_{3} · 3^{3} + O(3^{4}).

5. (a) Find the normalized Haar measure of the following subset of Q_{7}^{+}:

U = B (28, \frac{1}{50}) = \{x ∈ Q_{7} : |x − 28| < \frac{1}{50}\} .

(b) Find the normalized Haar measure of the subset Z_{7}^{∗} of Q_{7}^{∗}.

6. Suppose that K is a finite extension of Q_{p} and L is a finite extension of Q_{q}, with p ≠ q and assume that K and L have the same degree. Prove that there is a polynomial g ∈ Q[x] such that Q_{p}[x]/(g) ≅ K and Q_{q}[x]/(g) ≅ L. [Hint: Combine your solution to 3 with the weak approximation theorem.]