Defining polynomial
\(x^{8} - 19 x^{4} + 722\) ![]() |
Invariants
Base field: | $\Q_{19}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $4$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{19}(\sqrt{2})$ |
Root number: | $-1$ |
$|\Aut(K/\Q_{ 19 })|$: | $4$ |
This field is not Galois over $\Q_{19}.$ |
Intermediate fields
$\Q_{19}(\sqrt{2})$, 19.4.2.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{19}(\sqrt{2})$ $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{2} - x + 2 \) ![]() |
Relative Eisenstein polynomial: | \( x^{4} - 19 t \)$\ \in\Q_{19}(t)[x]$ ![]() |
Invariants of the Galois closure
Galois group: | $OD_{16}$ (as 8T7) |
Inertia group: | Intransitive group isomorphic to $C_4$ |
Unramified degree: | $4$ |
Tame degree: | $4$ |
Wild slopes: | None |
Galois mean slope: | $3/4$ |
Galois splitting model: | Not computed |