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

Question

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:

K := pAdicField(3, 100);
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

K := pAdicField(3, 100);
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);
InertiaDegree(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?

Answer 1

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:

K := pAdicField(3,30);
R<x> := PolynomialRing(K);
f := x^6+3*x^5-2;
L := LocalField(K,f);
RamificationIndex(L);
InertiaDegree(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:

K := pAdicField(3,30);
F<t> := UnramifiedExtension(K,2);
R<x> := PolynomialRing(F);
f := x^3+6*x^2+6*t*x+3*t;
L := ext<F|f>;
RamificationIndex(L,K);
InertiaDegree(L,K);
S<y> := PolynomialRing(L);
HasRoot(y^6+3*y^5-2);

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