Math 254A. Homework 11

Due Wednesday, 8 November

1. Prove that ℤ_{p} is compact.

2(nc). Exercise 7 on page 115.

3(nc). Determine the set \{|a|_{p} : a ∈ \overline{ℚ}_{p}\}. You may assume that the absolute value | · |_{p} on ℚ_{p} extends to a non-archimedean absolute value | · |_{p} on \overline{ℚ}_{p}.

4. Let K be a number field, let A be its ring of integers, let \overline{K} denote the algebraic closure of K (which is just \overline{ℚ}), and let \overline{A} be the integral closure of A in \overline{K}. Let 𝔭 be a nonzero prime of A. Show that there is a prime ideal 𝔮 of \overline{A} with 𝔮 ∩ A = 𝔭. Show also that 𝔮 defines in a natural way a valuation v: \overline{K} → ℚ ∪ \{∞\} such that v|_{K} = ord_{𝔭}, and that this valuation is not discrete.