Defining polynomial
\(x^{5} + 15 x^{4} + 5\)  |
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $5$ |
| Ramification exponent $e$: | $5$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $8$ |
| Discriminant root field: | $\Q_{5}(\sqrt{2})$ |
| 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: | \( x^{5} + 15 x^{4} + 5 \) |
Invariants of the Galois closure
| Galois group: | $F_5$ (as 5T3) |
| Inertia group: | $C_5$ |
| Unramified degree: | $4$ |
| Tame degree: | $1$ |
| Wild slopes: | [2] |
| Galois mean slope: | $8/5$ |
| Galois splitting model: | $x^{5} + 10 x^{3} - 60 x^{2} - 15 x - 148$ |