MATH 254 A: PROBLEM SET 3
MARTIN OLSSON
Due Wed Sep. 24
(1) Show by an example that a prime p Z can be totally ramied in two dierent number
elds K1 and K2 without being totally ramied in the compositum K1 K2 .
(2) Let K be a eld and | | be an absolute val
MATH 256 A: PROBLEM SET 1, DUE WED SEP 7
MARTIN OLSSON
(1) Let C be a category. Recall that for an object A C we dened a functor
hA : C op Set, B HomC (B, A).
Prove Yonedas lemma: The association A hA denes a fully faithful functor
h : C Fun(C op , Set).
MATH 254 A: PROBLEM SET 4
MARTIN OLSSON
Due Wed Oct 1
(1) Let K be a number eld, and let p Z be a prime. Show that there is a natural
isomorphism
K Q Qp
Kp ,
p
where on the right the product is taken over primes p OK and Kp denotes the p-adic
completion o
MATH 254 A: PROBLEM SET 6
MARTIN OLSSON
Due Wed Oct 22
(1) Here is another proof of quadratic reciprocity (the exercise consists of lling in the
details). Let p and q be distinct odd primes.
(a) Assume p 1 (mod 4), and q arbitrary. Then
p
q
= 1 x2 x +
1p
UC BERKELEY MATH 254A
PROBLEM SET 1
Last Updated: September 10, 2014. Please let me know if you nd any typos.
We will discuss these questions in class on Monday, Sept. 22nd.
(1) Questions about integral closure:
(a) Show that C[x, y]/(x2 y 3 ) is a domain
UC Berkeley Math 254A
Problem Set 3
November 12, 2014
Last Updated: November 12, 2014. Please let me know if you nd any typos.
We will discuss these questions in class on Mon. Nov. 24th.
Written solutions are due on Mon. Nov. 24nd.
1
Cyclotomic Fields
1.
UC Berkeley Math 254A
Problem Set 2
September 25, 2014
Last Updated: September 25, 2014. Please let me know if you nd any typos.
We will discuss these questions in class on Monday, Oct. 6th.
Written solutions are due on Wednesday, Oct. 8th.
1
The Minkowsk
MATH 254 A: PROBLEM SET 5
MARTIN OLSSON
Due Wed Oct 15
(1) Let G be a nite group and let : G C be a homomorphism. Show that
(g)
gG
is zero unless (g) = 1 for all g in which case the sum is equal to the order of G.
(2) Dene the n-th cyclotomic polynomial n
MATH 254 A: PROBLEM SET 8
MARTIN OLSSON
Due Fri Nov 21
(1) Let K = Q( d) with d a positive square free integer, and assume d is not congruent
to 1 mod 4. View K as a subeld of R by the standard embedding K R.
(a) Show that the group OK R>0 is isomorphic t
MATH 254A: PROBLEM SET 2
MARTIN OLSSON
Due Monday Sep. 15
(1) Let A be a Dedekind domain with eld of fractions K, and let p A be a prime. For
an ideal a dene (as in class)
a1 := cfw_x K|xa A.
Show that
a1 Ap = cfw_x K|x(a Ap ) Ap .
(2) Show that in Z[ 3]
MATH 254 A: PROBLEM SET 7
MARTIN OLSSON
Due Wed Nov 7
(1) Let K = Q(), where satises 5 + 1 = 0.
(a) Prove that OK = Z[].
(b) Using the Minkowski bound, show that the class number hK of K is 1.
(2) Let K = Q( d) where d is a square free integer.
(a) Show t
UC Berkeley Math 254A
Problem Set 3
October 10, 2014
Last Updated: October 10, 2014. Please let me know if you nd any typos.
We will discuss these questions in class on Wed. Oct. 22th.
Written solutions are due on Fri. Oct. 24nd.
1
Prime ideals vs. Discre