Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
The questions from the university of cambridge mathematical tripos part iii paper 28 local fields exam held on june 2, 2009. The paper includes questions on hensel's lemma, unique solutions of equations in local fields, continuity and mahler expansion of functions, unramified and ramified extensions, and the reciprocity map of local class field theory.
Typology: Exams
1 / 3
This page cannot be seen from the preview
Don't miss anything!
Tuesday, 2 June, 2009 1:30 pm to 4:30 pm
Attempt no more than FOUR questions. There are FIVE questions in total. The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
2
1 (i) State and prove a version of Hensel’s Lemma.
(ii) Let f (x) = x^3 − 3 x + 4. Show that the equation f (x) = 0 has a unique solution in Z 7 but no solutions in Z 5 or Z 3. Find how many solutions it has in Z 2.
(iii) Let f : Z 2 → Z 2 be the function defined by
f (x) =
1 if x ∈ 2 Z 2 − 1 if x /∈ 2 Z 2
Show that f is continuous and compute its Mahler expansion.
2 Let L/K be a finite extension of local fields. Show that L/K is unramified if and only if L = K(x) for some x ∈ oL whose minimal polynomial g satisfies g′(x) 6 ≡ 0 (mod mL).
Suppose that L/K is unramified, and M/K is arbitrary. Show that every kK - homomorphism kL → kM is induced by a unique K-homomorphism L → M. Deduce that L/K is Galois with cyclic Galois group.
Let L/K be a finite unramified extension, and M/K a totally ramified extension, both contained in a fixed algebraic closure of K. Show that the field LM is totally ramified over L.
Find an example of a finite extension N/Qp which is not of the form N = LM for an unramified extension L/Qp and a totally ramified extension M/Qp.
3 Let L/K be a finite extension of local fields. Show that L/K is totally ramified if and only if L = K(x) for some x ∈ oL whose minimal polynomial is an Eisenstein polynomial, and that in that case x is a uniformiser of L.
Show that if q is a power of p then Qp(ζq) is totally ramified. For what other values of n is it the case that Qp(ζn)/Qp is totally ramified?
If L/K is finite, Galois and totally ramified, define the ramification groups of L/K. Determine them for the extension Qp(ζq )/Qp.
4 (i) Prove that if L/K is an unramified extension of local fields of degree n then NL/K (L∗) = {x ∈ K∗^ | vK (x) ≡ 0 mod n}.
(ii) State and prove Hilbert’s Theorem 90. Deduce that if L/K is a finite unramified extension of local fields with Frobenius φL/K then for every x ∈ o∗ L with NL/K (x) = 1 there exists y ∈ o∗ L with φL/K (y)/y = x.
Part III, Paper 28
