Math 713 Problems

Problems are taken from Milne unless indicated otherwise. E denotes Exercise.

Submitting your work

Assignments should be written out and submitted on paper in the homework folder in my mailbox in LGRT by 5pm on the due date.

Assignments

1. Assigned 10 Sept 2023. Due 28 Sept 2023 in homework folder in my mailbox by 5pm.

   • E 2–1
   • E 2–3
   • E 2–7
   • E 2–8

2. Assigned 03 Oct 2023. Due 12 Oct 2023 in homework folder in my mailbox by 5pm.

   • E 3–1
   • E 3–2
   • E 3–4
   • (i) Show that there are only finitely many algebraic integers a of degree n, all of whose conjugates (including a) have absolute value 1. (ii) Show that any such algebraic integer is a root of unity.
   • Let F be the biquadratic field Q(sqrt(2),sqrt(3)) and let O_F be its ring of integers. Using software (for example gp, Sage, magma, …) determine a conjecture for how the ideal pO_F factors (i.e. what are e, f, g) in terms of p, for any rational prime p.

3. Assigned 20 Oct 2023. Due 31 Oct 2023 in homework folder in my mailbox by 5pm.

   • E 4–1
   • E 4–2
   • E 4–3
   • Compute class groups of the following imaginary quadratic fields (these are the discriminants): -19, -31, -47, -84, -163. (For the last one, cf. this for fun, especially Theorem 2.2.)
   • Compute class groups for the complex cubic fields of discriminants -31, -44, -59, and -283. (Hints: You can get polynomials from lmfdb.org by searching for these discriminants. Use Minkowski’s bound, not the one from class. You can use the gp-pari function bnfisprincipal to compute whether an ideal is principal. Here is an example of how it works. You can also use sage.)

4. Assigned 07 Nov 2023. Due 21 Nov 2023 in homework folder in my mailbox by 5pm.

   • E 5–1
   • E 5–2
   • E 5–3
   • Let F be a totally real number field of degree n. A unit u in O^\times is called totally positive if all its embeddings into R are positive. Let U_F be the subgroup of totally positive units. (a) Show that the index of U_F in O^\times is at most 2ⁿ. (b) Compute the index of U_F for the quadratic fields of discriminants 5, 8, 12, 13 and the cubic fields of discriminants 49, 81, 148, 169.
   • Consider the polynomials x⁵ - x³ - x² + x + 1, x⁵ - 2x⁴ + 2x³ - x² + 1, x⁵ - x⁴ + 2x³ - 4x² + x - 1, x⁵ - x⁴ - 4x³ + 3x² + 3x - 1, x⁵ - x⁴ + 2x² - 2x + 2. Each of them has a different Galois group. Experimentally determine which is which using the Chebetarov density theorem. (Hint: consider the conjugacy class of the identity.)