Defining polynomial
|
\(x^{8} + 52\)
|
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: | $1$ |
| $\card{ \Aut(K/\Q_{ 13 }) }$: | $4$ |
| This field is not Galois over $\Q_{13}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{13}(\sqrt{13})$, 13.4.3.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{13}$ |
| Relative Eisenstein polynomial: |
\( x^{8} + 52 \)
|