Defining polynomial
| \( x^{10} + 51 \) |
Invariants
| Base field: | $\Q_{17}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $10$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $9$ |
| Discriminant root field: | $\Q_{17}(\sqrt{17\cdot 3})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 17 })|$: | $2$ |
| This field is not Galois over $\Q_{17}.$ | |
Intermediate fields
| $\Q_{17}(\sqrt{17\cdot 3})$, 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: | $\Q_{17}$ |
| Relative Eisenstein polynomial: | \( x^{10} + 51 \) |
Invariants of the Galois closure
| Galois group: | $C_2\times F_5$ (as 10T5) |
| Inertia group: | $C_{10}$ |
| Unramified degree: | $4$ |
| Tame degree: | $10$ |
| Wild slopes: | None |
| Galois mean slope: | $9/10$ |
| Galois splitting model: | $x^{10} + 51$ |