Bill Allombert on Tue, 27 Jan 2015 20:37:49 +0100

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Re: Mixing variables in Mod expressions

On Tue, Jan 27, 2015 at 01:21:26PM -0600, Ariel Pacetti wrote:
> I have a somehow related problem/question to a computation I tried
> to do sometime ago (and never finished). Let say I have to elements
> a, b in two different number fields L,K (which are given as Mod(x,P)
> and Mod(y,Q)) and I want to know (the not well stated question) "for
> which rational primes they are congruent", meaning I want to know
> for which primes p, there exists a prime ideal dividing it in the
> composition of the two fields which divides the diference
> Mod(x,P)-Mod(y,Q).
> The naive answer is what was mentioned in the answer to John's
> question, just make a composition of the fields and compute the norm
> for example, but the composition might be huge (and hard to compute
> even when the two fields are of a manageable size, say 20 each).
> Then my question is "is there a better approach"?

Without restrictive hypothesis, there can be several compositum, each
of them giving a different answer to your question. Thus you need
a choice of compositum and embeddings.

If you ignore this problem, then you can compute the product of the
primes which divides at least one incarnation of Mod(x,P)-Mod(y,Q) using
resultants: just do