- Define the ring of integers of a number field. Give a
characterization of the ring of integers of quadratic and
cyclotomic extensions of
. Prove the result for quadratic
extensions.
- What is an integral basis? What is the norm of an ideal and why
is it finite?
- Define the terms ramification index and inertia degree. Which
primes ramify in quadratic and cyclotomic extensions of
?
- Let be an extension of number fields,
a
prime. Factor
as a product of primes
. What is the relation of the , and . If
is Galois, how does this affect things and why? Suppose
. How would you determine the ?
- Let
be a prime, and let
.
Determine all primes of
which ramify in . Determine
congruence conditions which describe all primes which split
(completely) in , and congruence conditions for those primes
which are inert.
- Let
be a prime,
, and
be the th cyclotomic extension. Show that
. Let be an odd prime which satisfies
for some odd positive integer . Determine the factorization
(type) of and . How do the primes which occur in the
factorization of factor in ? Work this out explicitly
with and .
- Let be a Galois extension of number fields,
a prime. Let
be a prime lying above
.
Describe the decomposition and inertia groups associated to
and
. Determine the order of the decomposition group. If is
another prime of lying above
, how are the decomposition
groups
and
related? Suppose that
is unramified in . Describe how to construct the Frobenius
map
. Show that for abelian
extensions, the Frobenius map depends only on
and not on .
- Let
,
, any prime of
lying above
and and the associated decomposition and
inertia groups. Let and be the associated fixed fields.
Determine them.
- Show that
has class number 2 using the
Minkowski bound.