Defining polynomial
|
\(x^{12} + 11\)
|
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $12$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $11$ |
| Discriminant root field: | $\Q_{11}(\sqrt{11})$ |
| Root number: | $-i$ |
| $\card{ \Aut(K/\Q_{ 11 }) }$: | $2$ |
| This field is not Galois over $\Q_{11}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{11}(\sqrt{11\cdot 2})$, 11.3.2.1, 11.4.3.1, 11.6.5.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{11}$ |
| Relative Eisenstein polynomial: |
\( x^{12} + 11 \)
|
Ramification polygon
| Residual polynomials: | $z^{11} + z^{10} + 1$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |