Defining polynomial
\(x^{7} - 7 x^{6} + 154\)  |
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $7$ |
| Ramification exponent $e$: | $7$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{7}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 7 })|$: | $7$ |
| This field is Galois and abelian over $\Q_{7}.$ | |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 7 }$. |
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}$ |
| Relative Eisenstein polynomial: | \( x^{7} - 7 x^{6} + 154 \) |
Invariants of the Galois closure
| Galois group: | $C_7$ (as 7T1) |
| Inertia group: | $C_7$ |
| Unramified degree: | $1$ |
| Tame degree: | $1$ |
| Wild slopes: | [2] |
| Galois mean slope: | $12/7$ |
| Galois splitting model: | $x^{7} - 305193 x^{5} - 17396001 x^{4} + 31813013127 x^{3} + 3355838747856 x^{2} - 1120814637257980 x - 161936091000794025$ |