Magma
Calculator
First steps
in Magma
Handbook Handbook
Number fields

Topics in number theory: Computer Lab

Friday April 30 and May 21, 9:00-13:00.

Examples.

In the lectures, I will go through various examples and more examples of how to use Magma for number theory. They can all be used in the Magma calculator

Assignment A.

Let K=Q(cube root of 2). Every prime p of Q splits into 1,2 or 3 primes of K. Here is a way to find which primes 2≤p<500 split completely:

R<x>:=PolynomialRing(Rationals());
K:=NumberField(x^3-2);
[p: p in PrimesUpTo(500) | #Decomposition(K,p) eq 3];

The first part of the assignment is to implement in Magma three other ways to generate the same list of primes. Note that Disc(OK)=-108=-2233 (use Discriminant(Integers(K))) so that 2 and 3 are the only ramified primes in K/Q.

Assignment B.

The second part of the assignment is to extend what we have done in the second lecture, and write a function in Magma that takes d and N and returns the list of cyclic extensions of Q of degree d and conductor N. (For d=3 and N=p prime this is done in class.)

function AbelianExtensions(N,d)
  ...
end function;

Submission.

Please submit your assignment (Magma code with maths in comments or as a separate document) by emailing it to me by May 25.

Questions/comments to Tim Dokchitser.