Defining polynomial
| \( x^{8} - 19 \) |
Invariants
| Base field: | $\Q_{19}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $8$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $7$ |
| Discriminant root field: | $\Q_{19}(\sqrt{19\cdot 2})$ |
| Root number: | $-i$ |
| $|\Aut(K/\Q_{ 19 })|$: | $2$ |
| This field is not Galois over $\Q_{19}.$ | |
Intermediate fields
| $\Q_{19}(\sqrt{19})$, 19.4.3.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{19}$ |
| Relative Eisenstein polynomial: | \( x^{8} - 19 \) |
Invariants of the Galois closure
| Galois group: | $SD_{16}$ (as 8T8) |
| Inertia group: | $C_8$ |
| Unramified degree: | $2$ |
| Tame degree: | $8$ |
| Wild slopes: | None |
| Galois mean slope: | $7/8$ |
| Galois splitting model: | Not computed |