Math 621  Problem Set #1  Due Wed., Jan. 30, 2002

1. Let M be the set of p-adic absolute values on ℚ (including the “usual” absolute value | · |_{∞}).

a) Let a_{1}, …, a_{n} ∈ ℚ and let v_{1}, …, v_{n} be distinct elements of M. Let ε_{1}, …, ε_{n} > 0. Show that there exists α ∈ ℚ such that |α − a_{i}|_{v_{i}} < ε_{i} for all i. (Hint: Mimic the ususal proof of the Chinese Remainder Theorem.)

b) Find an α ∈ ℚ such that |α − 2|_{3} < \frac{1}{3} and |α − 1|_{5} < 1.

c) Find an α ∈ ℚ such that |α − 2|_{3} < \frac{1}{3} and |α − 1|_{∞} < 1.

d) What are all the solutions to the system in (b)? to the system in (c)?

2. a) Let u be a unit in ℤ_{2}. Show that u is a square if and only if u ≡ 1(mod 8). b) Let p, q be primes. Show that:

(i) q is not a square in ℤ_{p} if p ≠ 2, q and (\frac{q}{p}) = −1 (Legendre symbol); (ii) q is a square in ℤ_{p} if p ≠ 2, q and (\frac{q}{p}) = 1; (iii) if p = 2 ≠ q, then q is a square in ℤ_{p} if and only if q ≡ 1 (mod 8); (iv) q is not a square in ℤ_{p} if p = q.

3. a) Prove that ℤ_{p} is compact. (Hint: Consider the map ℤ_{p} → ℤ/p × ℤ/p^{2} × ⋯ and use Tychonoff’s Theorem.)

b) Deduce that ℚ_{p} is locally compact. Is it compact?

c) Show that ℤ_{p} and ℚ_{p} are each totally disconnected.

4. Let R = ℤ_{2}[ζ_{3}, ^{3} \sqrt{2}, \sqrt{2}], where ζ_{3} is a primitive cube root of unity. Let K be the fraction field of R.

a) Is K Galois over ℚ_{2}? If so, find its Galois group.

b) What is the ramification index of this extension? What is the residue class degree?

c) Find the maximal unramified subextension, and also the maximal tamely ramified subextension.

5. Let K be a complete discrete valuation field.

a) Let K ⊂ L ⊂ E be finite separable field extensions. Show that E is unramified over K iff L is unramified over K and E is unramified over L.

b) Let K′, L be finite separable field extensions of K, and let L′ be their compositum (inside some algebraic closure of K). Show that if L is unramified over K then L′ is unramified over K′.

c) Under the hypotheses of (b), show that if K′, L are each unramified over K, then so is L′.

d) Redo parts (a)-(c) with “unramified” replaced by “tamely ramified”.