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: |
\( x^{8} + 52 \)
|
Ramification polygon
Residual polynomials: | $z^{7} + 8z^{6} + 2z^{5} + 4z^{4} + 5z^{3} + 4z^{2} + 2z + 8$ |
Associated inertia: | $2$ |
Indices of inseparability: | $[0]$ |