Orals Questions:

Algebraic Number Theory

- For which integers n is 3500 a square mod n?
  - Does the method you used to determine this work in general?
  - Is this effectively (polynomial-time) computable with the method you used?
(In my case, Legendre symbol.)
    What about in general? (Think Jacobi symbol.)

- Regarding the field extension K := Q(i) of Q:
  - Give the conductor.
  - Describe the splitting/ramification of primes (including infinite).
  - Relate the Galois group to a ray class group via Artin map.
  - Factor \zeta_K as a product of \zeta_Q and another function (directly from
definition of \zeta).
  - Identify this other function; relate it to the above; where does it converge
as a Dirichlet series?
    * Also related this to: how do we count the number of ideals in a ring of
integers with norm bounded by n?


Algebraic Topology

- Compute cohomology group and ring structure on CP^n.
- Is there an orientation-reversing diffeomorphism on CP^2?