Math 254A. Homework 14

Due Thursday, 7 December

1. Determine (explicitly) all quadratic extensions of ℚ_{2} . (You may use Kummer theory, but don’t have to.) Determine e and f for each of these, as well as e and f over ℚ_{2} for the compositum of each pair of quadratic extensions.

2. Exercise 3 on page 166.

3. Let L/K be a finite extension of fields, let v be a non-archimedean valuation of K, let A be the valuation ring of v, and let B be the integral closure of A in L. For extensions w of v to L, let B_{w} denote the valuation ring of w.

In parts (a) and (b), do not assume that K is complete, that v is discrete, or that B is finite over A.

(a). Use results from Section 8 of Chapter II to show that

B = ⋂_{w} B_{w} ,

where w ranges over all extensions of v to L. (b). For all w as in part (a), show that

B_{w} = B_{𝔓} ,

where 𝔓 = {α ∈ B : w(α) > 0}. (This is Exercise 6 on page 166.) (c). Use the result of part (b) to show that if A, K, L, and B are as in the “usual picture,” and if | · | is an absolute value on K coming from a prime ideal 𝔭 ≠ 0 of A, then all extensions of | · | to L come from prime ideals 𝔮 of B lying over 𝔭.