Defining polynomial
\(x^{9} + 26\) |
Invariants
Base field: | $\Q_{13}$ |
Degree $d$: | $9$ |
Ramification exponent $e$: | $9$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $8$ |
Discriminant root field: | $\Q_{13}$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 13 }) }$: | $3$ |
This field is not Galois over $\Q_{13}.$ | |
Visible slopes: | None |
Intermediate fields
13.3.2.1 |
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: | \( x^{9} + 26 \) |
Ramification polygon
Residual polynomials: | $z^{8} + 9z^{7} + 10z^{6} + 6z^{5} + 9z^{4} + 9z^{3} + 6z^{2} + 10z + 9$ |
Associated inertia: | $3$ |
Indices of inseparability: | $[0]$ |