Defining polynomial
\(x^{8} + 712860 x^{4} - 1295029 x^{2} + 127042344900\)  |
Invariants
| Base field: | $\Q_{109}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $4$ |
| Discriminant root field: | $\Q_{109}$ |
| Root number: | $-1$ |
| $|\Gal(K/\Q_{ 109 })|$: | $8$ |
| This field is Galois and abelian over $\Q_{109}.$ | |
Intermediate fields
| $\Q_{109}(\sqrt{2})$, $\Q_{109}(\sqrt{109})$, $\Q_{109}(\sqrt{109\cdot 2})$, 109.4.0.1, 109.4.2.1, 109.4.2.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 109.4.0.1 $\cong \Q_{109}(t)$ where $t$ is a root of \( x^{4} - x + 30 \) |
| Relative Eisenstein polynomial: | \( x^{2} - 109 t^{2} \)$\ \in\Q_{109}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_2\times C_4$ (as 8T2) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Unramified degree: | $4$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | Not computed |