Algebraic Number Theory
Math 514A Fall 2007
Problem Set 8
Due: Tuesday, Oct. 30th
1. a) Explicitly find the unramified extension of Q7of degree 3. Also find a tamely
ramified Galois extension of Q7of degree 3.
b) Do the same with Q7replaced by F7((t)).
2. Let R=Z2[ζ3,3
√2,√2], where ζ3is a primitive cube root of unity. Let Kbe the
fraction field of R.
a) Is KGalois over Q2? If so, find its Galois group.
b) What is the ramification index of this extension? What is the residue class
degree?
c) Find the maximal unramified subextension, and also the maximal tamely
ramified subextension.
3. Describe the higher ramification groups at the primes over 2 and 3 in K/Q, where
K=Q(ζ3, α) with α3= 2.
4. Describe the higher ramification groups at the primes over pin K/Q, where K=
Q(ζp2) is the p2th cyclotomic field.
5. Let ζp2be a primitive p2th root of unity. Describe the higher ramification groups
at the primes over pin K=Qp(ζp2) over Qp; And use these higher ramification
groups to compute the discriminant δK.