Defining polynomial
\( x^{5} + 15 x^{2} + 5 \) |
Invariants
Base field: | $\Q_{5}$ |
Degree $d$: | $5$ |
Ramification exponent $e$: | $5$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{5}$ |
Root number: | $1$ |
$|\Aut(K/\Q_{ 5 })|$: | $1$ |
This field is not Galois over $\Q_{5}.$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 5 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{5}$ |
Relative Eisenstein polynomial: | \( x^{5} + 15 x^{2} + 5 \) |
Invariants of the Galois closure
Galois group: | $D_5$ (as 5T2) |
Inertia group: | $D_{5}$ |
Unramified degree: | $1$ |
Tame degree: | $2$ |
Wild slopes: | [3/2] |
Galois mean slope: | $13/10$ |
Galois splitting model: | $x^{5} - 5 x^{3} + 10 x - 4$ |