Defining polynomial
|
\(x^{11} + 13\)
|
Invariants
| Base field: | $\Q_{13}$ |
| Degree $d$: | $11$ |
| Ramification exponent $e$: | $11$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $10$ |
| Discriminant root field: | $\Q_{13}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 13 }) }$: | $1$ |
| This field is not Galois over $\Q_{13}.$ | |
| Visible slopes: | None |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 13 }$. |
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{13}$ |
| Relative Eisenstein polynomial: |
\( x^{11} + 13 \)
|
Ramification polygon
| Residual polynomials: | $z^{10} + 11z^{9} + 3z^{8} + 9z^{7} + 5z^{6} + 7z^{5} + 7z^{4} + 5z^{3} + 9z^{2} + 3z + 11$ |
| Associated inertia: | $10$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $F_{11}$ (as 11T4) |
| Inertia group: | $C_{11}$ (as 11T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $10$ |
| Tame degree: | $11$ |
| Wild slopes: | None |
| Galois mean slope: | $10/11$ |
| Galois splitting model: |
$x^{11} - 13$
|