Defining polynomial
| \( x^{7} + 7499223 \) |
Invariants
| Base field: | $\Q_{127}$ |
| Degree $d$ : | $7$ |
| Ramification exponent $e$ : | $7$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $6$ |
| Discriminant root field: | $\Q_{127}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 127 })|$: | $7$ |
| This field is Galois and abelian over $\Q_{127}$. | |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 127 }$. |
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{127}$ |
| Relative Eisenstein polynomial: | \( x^{7} + 7499223 \) |
Invariants of the Galois closure
| Galois group: | $C_7$ (as 7T1) |
| Inertia group: | $C_7$ |
| Unramified degree: | $1$ |
| Tame degree: | $7$ |
| Wild slopes: | None |
| Galois mean slope: | $6/7$ |
| Galois splitting model: | Not computed |