[Solved] Actuarial Science

Assignment Prompt & Instructions

• 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?
• Let K = Q(α), where Irrα,Q(x) = x
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?
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