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)) Bjorn poonen@math.berkeley.edu