Defining polynomial
\( x^{10} - 17 x^{5} + 867 \) |
Invariants
Base field: | $\Q_{17}$ |
Degree $d$: | $10$ |
Ramification exponent $e$: | $5$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $8$ |
Discriminant root field: | $\Q_{17}(\sqrt{3})$ |
Root number: | $1$ |
$|\Aut(K/\Q_{ 17 })|$: | $2$ |
This field is not Galois over $\Q_{17}.$ |
Intermediate fields
$\Q_{17}(\sqrt{3})$, 17.5.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{17}(\sqrt{3})$ $\cong \Q_{17}(t)$ where $t$ is a root of \( x^{2} - x + 3 \) |
Relative Eisenstein polynomial: | $ x^{5} - 17 t \in\Q_{17}(t)[x]$ |
Invariants of the Galois closure
Galois group: | $F_5$ (as 10T4) |
Inertia group: | Intransitive group isomorphic to $C_5$ |
Unramified degree: | $4$ |
Tame degree: | $5$ |
Wild slopes: | None |
Galois mean slope: | $4/5$ |
Galois splitting model: | Not computed |