Topics in number theory: Computer Lab

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(O_K)=-108=-2²3³ (use Discriminant(Integers(K))) so that 2 and 3 are the only ramified primes in K/Q.

• 1. Explain why p splits completely in K/Q if and only if p=1 mod 3 and 2^(p-1)/3=1 in F_p=Z/pZ. Implement this criterion.
  
  
• 2. Use the Kummer-Dedekind factorisation theorem for the defining polynomial x³-2 for K over Q.
  
  
• 3. Show that p splits completely in K/Q if and only if p=1 mod 3 and a prime ideal P=(a+bζ₃) of Q(ζ₃) above p splits completely in the abelian extension K(ζ₃)/Q(ζ₃).
  
  
• 4. (Optional) If you know about ray class groups, show that P splits completely in K(ζ₃)/Q(ζ₃) if and only if it has a generator which is 1 mod m, where the modulus m = 6 Z[ζ₃]. Implement this criterion.
  
  

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.