# Global Theory

1. 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.

2. What is an integral basis? What is the norm of an ideal and why is it finite?

3. Define the terms ramification index and inertia degree. Which primes ramify in quadratic and cyclotomic extensions of ?

4. 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 ?

5. 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.

6. 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 .

7. 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 .

8. Let , , any prime of lying above and and the associated decomposition and inertia groups. Let and be the associated fixed fields. Determine them.

9. Show that has class number 2 using the Minkowski bound.

root 2007-06-06