Defining polynomial
\(x^{9} + 11\) |
Invariants
Base field: | $\Q_{11}$ |
Degree $d$: | $9$ |
Ramification exponent $e$: | $9$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $8$ |
Discriminant root field: | $\Q_{11}$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 11 }) }$: | $1$ |
This field is not Galois over $\Q_{11}.$ | |
Visible slopes: | None |
Intermediate fields
11.3.2.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{11}$ |
Relative Eisenstein polynomial: | \( x^{9} + 11 \) |
Ramification polygon
Residual polynomials: | $z^{8} + 9z^{7} + 3z^{6} + 7z^{5} + 5z^{4} + 5z^{3} + 7z^{2} + 3z + 9$ |
Associated inertia: | $6$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_9:C_6$ (as 9T10) |
Inertia group: | $C_9$ (as 9T1) |
Wild inertia group: | $C_1$ |
Unramified degree: | $6$ |
Tame degree: | $9$ |
Wild slopes: | None |
Galois mean slope: | $8/9$ |
Galois splitting model: | $x^{9} - 11$ |