On Ordering for Drew Sutherland

[R. Andrew Ohana]

Hey William (+anyone else who is interested),

I've started playing around with generalizing the ordering you described on Friday for general number fields (the one which orders primes based on norm then lexigraphically by the polynomial/element P with coefficients in [0,p-1] such (p,P) generate the prime). The idea I've had to generalize this to arbitrary ideals is to order first by norm, and then lexigraphically by a particular polynomial so that (n,P) generates the ideal. The question is what that polynomial should be, and I feel like I may have an answer. The idea stems from Hermite normal forms, and given by the below statements that at least appear true from my computations. If anyone knows any of them to be either true or false or sees a quick proof one way or the other, please let me know.

In all statements, assume that K is an arbitrary number field of degree n, with a chosen integral basis, that I is an ideal of O_K and that H is the Hermite normal form attached to I for our given choice of integral basis. Additionally, we are assume for the purpose of these statements that Hermite normal forms are upper triangular.

1) If i > j then H_{i,i} <= H_{j,j} (or even better that H_{i,i} | H_{j,j} )

2) There exists some k such that H_{k,k} = gcd(H) = gcd{H_{i,j} : 0 < i,j <= n }

3) If k = min{i : H_{i,i} = gcd(H) } then I = < N(I), P_k >, where P_i is the polynomial with coefficients given by the i-th column of H

I seem to recall learning the first statement was true at some point, but I don't remember for sure. It isn't needed to define the polynomial that we want, but I believe that it will be needed in proving 2 and/or 3 (supposing they are in fact true statements).

For prime ideals, these statements appear to give the same polynomial one would obtain through factoring. Assuming this works out, I think it might be good to have our lexigraphical ordering start from the zeroeth coefficient of the polynomial rather than the nth (i.e. in the fashion of writing down number field elements as opposed to writing down polynomials).

--

Andrew

[R. Andrew Ohana]

I got 1) and 2) by showing

a) H_{i,i} | H_{j,j} for i > j
b) H_{i,i} = gcd(H_i) where H_i is the i-th column of H.