Defining polynomial
| \( x^{8} - 19 x^{4} + 5776 \) |
Invariants
| Base field: | $\Q_{19}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $6$ |
| Discriminant root field: | $\Q_{19}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 19 })|$: | $8$ |
| This field is Galois over $\Q_{19}.$ | |
Intermediate fields
| $\Q_{19}(\sqrt{2})$, $\Q_{19}(\sqrt{19})$, $\Q_{19}(\sqrt{19\cdot 2})$, 19.4.2.1, 19.4.3.1 x2, 19.4.3.2 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{19}(\sqrt{2})$ $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{2} - x + 2 \) |
| Relative Eisenstein polynomial: | $ x^{4} - 19 t^{4} \in\Q_{19}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $D_4$ (as 8T4) |
| Inertia group: | Intransitive group isomorphic to $C_4$ |
| Unramified degree: | $2$ |
| Tame degree: | $4$ |
| Wild slopes: | None |
| Galois mean slope: | $3/4$ |
| Galois splitting model: | Not computed |