deciding whether two padic extensions are isomorphic

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

-- 
=============================================================
Fernando Q. Gouveahttp://www.colby.edu/~fqgouvea
Carter Professor of Mathematics
Dept. of Mathematics
Colby College
5836 Mayflower Hill
Waterville, ME 04901

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