Bjorn Poonen on Tue, 29 Sep 1998 17:31:19 -0700 (PDT)

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Re: Computations in extensions of Q_p

Dear Roland,

You wrote

>I'd like to do some simple arithmetic with elements of unramified
>extensions of p-adic fields.  In other words, I'd like to compute with
>power series in "p" whose coefficients lie in an finite field with p^a
>elements, for a>1.  Is this easy/possible with PARI?

When I have needed the degree d unramified extension K of Q_p in PARI,
my approach has been to
1) use ffinit to pick an irreducible polynomial of degree d over F_p,
2) lift it to a monic polynomial in Z[x],
3) multiply it by Mod(1,p^n) for some fixed integer n to get
	a polynomial f(x) in (Z/p^n)[x], and then
4) do all my computations in the ring (Z/p^n)[x] / (f(x)),
	which is isomorphic to O_K/(p^n).

This doesn't exactly solve the problem, but it has worked reasonably well
for me.  I suppose it depends on what you want to do.
A typical element of this ring might look like

Mod(Mod(2, 729)*x^2 + Mod(1, 729)*x + Mod(728, 729), Mod(1, 729)*x^5 + Mod(2, 729)*x + Mod(1, 729))