Defining polynomial
\(x^{8} - 153\) ![]() |
Invariants
Base field: | $\Q_{17}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $8$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $7$ |
Discriminant root field: | $\Q_{17}(\sqrt{17})$ |
Root number: | $1$ |
$|\Gal(K/\Q_{ 17 })|$: | $8$ |
This field is Galois and abelian over $\Q_{17}.$ |
Intermediate fields
$\Q_{17}(\sqrt{17})$, 17.4.3.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{17}$ |
Relative Eisenstein polynomial: | \( x^{8} - 153 \) ![]() |
Invariants of the Galois closure
Galois group: | $C_8$ (as 8T1) |
Inertia group: | $C_8$ |
Unramified degree: | $1$ |
Tame degree: | $8$ |
Wild slopes: | None |
Galois mean slope: | $7/8$ |
Galois splitting model: | Not computed |