Defining polynomial
\(x^{4} - 7 x^{2} + 147\)  |
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $4$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $2$ |
| Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
| Root number: | $-1$ |
| $|\Gal(K/\Q_{ 7 })|$: | $4$ |
| This field is Galois and abelian over $\Q_{7}.$ | |
Intermediate fields
| $\Q_{7}(\sqrt{3})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}(\sqrt{3})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} - x + 3 \) |
| Relative Eisenstein polynomial: | \( x^{2} - 7 t \)$\ \in\Q_{7}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_4$ (as 4T1) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Unramified degree: | $2$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | $x^{4} - x^{3} - 9 x^{2} + 9 x + 11$ |