deciding whether two padic extensions are isomorphic

[Fernando Gouvea <fqgouvea@colby.edu>]

In my book on the p-adic numbers, I mention the GP command padicfields, which lists out the (finitely many) extensions of a given Q_p of a given degree. With the flag 1, it lists the polynomial that generates the extension, followed by the ramification index e, the residue degree f, the (power of 3 in) the discriminant, and the number of different embeddings in an algebraic closure.

gp > padicfields(3,4,1)
%14 = [[x^4 + 13*x^3 + 64*x^2 + 61*x + 40, 1, 4, 0, 1], 
       [x^4 + 2*x^3 + 11*x^2 + 10*x + 4, 2, 2, 2, 1], 
       [x^4 + 2*x^3 + 8*x^2 + 13*x + 7, 2, 2, 2, 1], 
       [x^4 + 3, 4, 1, 3, 2], 
       [x^4 + 6, 4, 1, 3, 2]]

Earlier in the book I had introduced a field F obtained from Q_3 by adjoining a cube root of 1 and a square root of 2. That is an extension of degree 4 with e=f=2, so it is either the second or the third in this list. How might one decide which? In other words, given two polynomials of degree 4, is there a way to use GP to decide whether they define the same extension?

A similar question might be asked about the two totally ramified extensions as well, since their invariants are also the same.

Thanks,

Fernando

-- 
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Fernando Q. Gouvea         http://www.colby.edu/~fqgouvea
Carter Professor of Mathematics
Dept. of Mathematics
Colby College              
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