Defining polynomial
| \( x^{10} - 7 x^{5} + 147 \) |
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$ : | $10$ |
| Ramification exponent $e$ : | $5$ |
| Residue field degree $f$ : | $2$ |
| Discriminant exponent $c$ : | $8$ |
| Discriminant root field: | $\Q_{7}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 7 })|$: | $2$ |
| This field is not Galois over $\Q_{7}$. | |
Intermediate fields
| $\Q_{7}(\sqrt{*})$, 7.5.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}(\sqrt{*})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} - x + 3 \) |
| Relative Eisenstein polynomial: | $ x^{5} - 7 t \in\Q_{7}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $F_5$ (as 10T4) |
| Inertia group: | Intransitive group isomorphic to $C_5$ |
| Unramified degree: | $4$ |
| Tame degree: | $5$ |
| Wild slopes: | None |
| Galois mean slope: | $4/5$ |
| Galois splitting model: | $x^{10} - x^{8} + 6 x^{6} - 11 x^{4} + 6 x^{2} - 5$ |