Math 254A. Homework 13

Due Tuesday, 28 November

1(nc). Let K be a complete discretely valued field, and fix an algebraic closure \overline{K} of K

(which is a valued field, but is not discretely valued, and may not be complete).

(a). Let L and L′ be subfields of \overline{K}, finite and unramified over K. Let λ and λ′ be the residue fields of L and L′, respectively. Show that if λ ⊆ λ′ then L ⊆ L′.

(b). Let L_{1} and L_{2} be subfields of \overline{K}, finite and unramified over K. Let λ_{1} , λ_{2} , and λ_{3} be the residue fields of L_{1} , L_{2} , and L_{1}L_{2} , respectively. Show that λ_{1}λ_{2} = λ_{3} . (In other words, show that the residue field of the compositum of two finite unramified extension fields of K in some common larger field is the compositum of the residue fields.)

2. Let L/K be a finite extension of complete discretely valued fields, whose residue field extension is separable. Let A and B be the valuation rings of K and L, respectively. Without relying on Exercise 2 on page 52, show that there is a θ ∈ B such that B = A[θ] .

3. Show that an infinite algebraic extension of ℚ_{p} is never complete (cf. Exercise 1 on page 134). [Hint: Use the last question on Homework 12.]