Defining polynomial
\(x^{4} - x + 11\)  |
Invariants
| Base field: | $\Q_{71}$ |
| Degree $d$: | $4$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{71}(\sqrt{7})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 71 })|$: | $4$ |
| This field is Galois and abelian over $\Q_{71}.$ | |
Intermediate fields
| $\Q_{71}(\sqrt{7})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 71.4.0.1 $\cong \Q_{71}(t)$ where $t$ is a root of \( x^{4} - x + 11 \) |
| Relative Eisenstein polynomial: | \( x - 71 \)$\ \in\Q_{71}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_4$ (as 4T1) |
| Inertia group: | trivial |
| Unramified degree: | $4$ |
| Tame degree: | $1$ |
| Wild slopes: | None |
| Galois mean slope: | $0$ |
| Galois splitting model: | $x^{4} - x^{3} + 2 x^{2} + 4 x + 3$ |