# Implementation in MAGMA: Field extension over the p-adics with a polynomial which is neither inertial nor Eisenstein

Let $$K = Q_3$$ and $$L = K(a)$$ be the extension of $$K$$ defined by the polynomial $$f = x^6+3x^5-2$$ (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 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?

It looks like you have two different polynomials $$f$$. The original is $$f=x^6+3x^5-2$$. The one you typed in your second Magma session was $$f=x^6-x^4-5$$.

There are at least two ways to define your extension in Magma. One involves using the "LocalField" command:

R<x> := PolynomialRing(K);
f := x^6+3*x^5-2;
L := LocalField(K,f);
RamificationIndex(L);

Another way is to build the extension as a totally ramified cubic extension of the unramified quadratic extension, using the defining polynomials listed in the LMFDB link you gave:

The last command shows your original polynomial $$f$$ has a root in the extension $$L$$, and thus is a defining polynomial for $$L$$.