Actuarial Science

• Prove that the class group of a number field is finite. 

• What is your favorite proof of quadratic reciprocity? 

• Prove that there exists a field of order p

n for every prime p and positive integer n.

[Peres (Stat)]

• Prove that Q[

3] has no unramified extensions. [Ribet]

• Describe the ring of integers in Q(ζp∞). [Coleman]

• Tell me about integral extensions. [Hartshorne] item• 

What is a Dedekind domain?

[Hartshorne]

• An example of a domain which is noetherian, integrally closed, and not one-

dimensional.

• An example of a domain which is integrally closed, one-dimensional, and not

noetherian.

• State Leopoldt’s conjecture. [Coleman]

• State the main theorem of Class Field Theory in terms of id`eles. [Coleman]

• Define id`eles.

• Give the Class Field Theory correspondence.

• What subgroup corresponds to the kernel of the Artin map for unramified exten-

sions?

• Describe the maximal abelian extension L of Q unramified outside p explicitly,

using the id`elic formulation of Class Field Theory.

• Show that Gal(L/Q) ∼= Z

× directly, from the Artin map on id`eles.

• Why are elliptic curves important in number theory?

• Say everything you can about Q(

√

−5): ring of integers, discriminant, which primes

ramify, split or remain inert, and whether Z[

√

−5] is a PID. What is the class number?

3 + 2x + 1. 

What is DK (the discriminant)?

Which primes ramify in K ? What is the splitting behavior of 2 and of 3?

• Let L = Q(α1, α2, α3) be the splitting field of x

3 + 2x + 1 over Q. (i.e., αi are the

roots). 

. What is Gal(L/Q)? How does 59 ramify in K = Q(α1)? In L?