18.786 | Spring 2010 | Graduate

# Topics in Algebraic Number Theory

## Assignments

The problem sets for this course account for 50% of each student’s grade. Students are encouraged to work on the homework in groups, but you must write up your own solutions.

ASSIGNMENTS HINTS
Problem set 1 (PDF)
Problem set 2 (PDF)

- In problem 5, take an arbitrary element a + bu + cu^2 of Q (u) where u^3 = 2, with a,b,c in Q, and compute its characteristic polynomial from the definition.

- In problem 6, to produce an element of a quadratic subextension of the cyclotomic field, add the elements of an H-orbit of a suitable generator of the cyclotomic field, where H is a suitable subgroup of the Galois group G.

- In problem 7, if you haven’t seen tensor products of modules before, see an algebra book (e.g. Lang or Hungerford), or the Wikipedia page.

Problem set 3 (PDF)
Problem set 4 (PDF)
Problem set 5 (PDF)
Problem set 6 (PDF) Two fixes: In problem 1) assume the ideal \$\frak{a}\$ is prime. In problem 6 you have to show U/U_1 is cyclic, not U/U_0 (which is trivial). A non-archimedean local field in the problems (and in this course) will always be a finite extension of \$\mathbb{Q}_p\$.
Problem set 7 (PDF)
Problem set 8 (PDF)
Problem set 9 (PDF)
Problem set 10 (PDF)
Problem set 11 (PDF)

- For problem 6, the following gp functions may come in handy: polcompositum(f,g) returns polynomials defining the compositum(s) of the fields defined by the irreducible polynomials f and g. For F = nfinit(f) a number field, F.disc returns the discriminant of F.

- For problem 7, you may use gp to do factoring mod p, as usual.

Take-home final (PDF) Problem 11: use a theorem we used early in class field theory which provides generators for a Galois group over Q. Also, see pg 95-96 of Samuel.

Spring 2010
Problem Sets