# Doing a field extension over the p-adics with a polynomial which is neither inertial nor Eisenstein

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### Magma Newbie

Jan 20, 2021, 10:53:28 AMJan 20
to MAGMA User
Dear all,

since I am new here, I would like to apologize for the mistakes I make here in advance!
Here is the issue I have: Let K = Q_3 and L = K(a) be the extension of K defined by the polynomial f = x^6+3*x^5-2 (i.e. this is the minimal polynomial of a over K). Now I would like to obtain this field L in Magma but I have issues with that.

More precisely, it seems like that I can only define extensions over the p-adics by giving either an inertial or an Eisenstein polynomial, and f is neither inertial nor Eisenstein. The code I tried is this one:

R<x> := PolynomialRing(K);
L := ext< K | x^6+3*x^5-2 >;

which gives me the error message:

>> L := ext< K | x^6+3*x^5-2 >;
^
Runtime error in ext< ... >: Polynomial must be Eisenstein or inertial

I also tried to work around that by defining the maximal unramified subextension F/K (which has degree 2, see https://www.lmfdb.org/LocalNumberField/3.6.6.1) like this

F := UnramifiedExtension(K,2);
R<x> := PolynomialRing(F);
f := x^6-x^4-5;
f_factor := Factorization(f)[1][1];
L := ext< F | f_factor >;
Degree(L,K);
RamificationIndex(L,K);

But in the end, Magma says that my extension L/K is unramified of degree 6 which cannot be true (cf. the LMFDB page where it says that the extension has inertial degree 2 instead of 6).

Could you please tell me if there is a way around this problem or what I did wrong?

Thank you!