Math 514b Algebraic Number Theory Spring 1996

Homework assignment 2:

Until further notice, a “non-archimedean local field” will be the fraction field of a complete DVR with perfect residue field. In these exercise, we will drop the words “non-archimedean.”

8. Let L/K be a separable finite extension of local fields. Prove that there exists an x ∈ O_{L} such that O_{L} = O_{K}[x]. (Hints: Consider the unramified and totally ramified cases first. Then show that there exists x ∈ O_{L} whose residue class is a primitive element for the residue field extension and a monic polynomial R ∈ O_{K} such that R(x) is a uniformiser of O_{L}.)

9. Describe, for p = 2, 3, and 5, the extension of Q_{p} obtained by adjoining a root of x^{2} − 2. Compute the ramification groups, the different, and the discriminant.

10. Let f = x^{3} − x + 1. For p = 2, 5, 23, and 59 find the splitting behavior of f over Q_{p}. If f has an irreducible factor of degree > 1, find the ramification and residue degrees of the extension obtained by adjoining a root of the factor. Compare your results with facts from last semester.

11. Suppose L/K is a Galois extension of local fields with group G and H ⊆ G is a normal subgroup. Thus K′ = L^{H} is Galois over K with group G/H. Prove that the ramification groups of K′/K satisfy

(G/H)_{i} = G_{i}H/H

for i = 0 and 1. This is false for i > 2; can you find an example?

12. Prove that x^{p} − x − p^{−1} is irreducible over Q_{p} and that the extension obtained by adjoining a root is Galois. If G is the Galois group, show that G_{1} = G and G_{2} = 0.

13. If K is a local field and L/K is a finite algebraic extension, the absolute value on K has a unique extension to L and L is complete with respect to the extended absolute value. In this exercise you will show that an infinite algebraic extension of K is never complete with respect to any absolute value extending that of K.

a) Show that if a is algebraic over K of degree d there there exists a positive integer N such that a does not satisfy any congruence

∑_{i=0}^{d−1} c_{i}a^{i} ≡ 0 (mod π^{N}O_{K(a)})

where c_{i} ∈ O_{K} and not all c_{i} are divisible by π. (Here π is a uniformiser of O_{K}.)

b) Suppose

K ⊆ K_{1} ⊆ K_{2} ⊆ …

is a sequence of finite separable subextensions of L with K_{i} ≠ K_{i+1} and let L′ = ∪K_{i}. Cook up a Cauchy sequence in L′ which, if it had a limit, would converge to something transcendental over K. (Hint: Let

a_{n} = ∑_{j=0}^{n} b_{j}π^{N_{j}}

where b_{j} ∈ O_{K_{i}} is a primitive element for K_{i}/K and the N_{j} are chosen suitably.)

c) Conclude that L is not complete if the separable closure of K in L has infinite degree over K. If this degree is finite, then L contains an infinite purely inseparable extension of K; show that in this case too, L is not complete.

14. (Optional) Describe the algebraic closure and the completion of the algebraic closure of the (complete) field of Laurent series C((t)), where C is the complex numbers.