Defining polynomial
\(x^{8} + 13\)
|
Invariants
Base field: | $\Q_{13}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $8$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $7$ |
Discriminant root field: | $\Q_{13}(\sqrt{13})$ |
Root number: | $1$ |
$\Aut(K/\Q_{13})$: | $C_4$ |
This field is not Galois over $\Q_{13}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{13}(\sqrt{13})$, 13.4.3.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{13}$ |
Relative Eisenstein polynomial: |
\( x^{8} + 13 \)
|
Ramification polygon
Not computedInvariants of the Galois closure
Galois group: | $\OD_{16}$ (as 8T7) |
Inertia group: | $C_8$ (as 8T1) |
Wild inertia group: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $8$ |
Wild slopes: | None |
Galois mean slope: | $7/8$ |
Galois splitting model: | $x^{8} + 26 x^{6} + 65 x^{4} + 52 x^{2} + 13$ |