Defining polynomial
| \( x^{11} + 88 x^{10} + 11 \) |
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$ : | $11$ |
| Ramification exponent $e$ : | $11$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $20$ |
| Discriminant root field: | $\Q_{11}$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 11 })|$: | $1$ |
| This field is not Galois over $\Q_{11}$. | |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 11 }$. |
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{11}$ |
| Relative Eisenstein polynomial: | \( x^{11} + 88 x^{10} + 11 \) |
Invariants of the Galois closure
| Galois group: | $C_{11}:C_5$ (as 11T3) |
| Inertia group: | $C_{11}$ |
| Unramified degree: | $5$ |
| Tame degree: | $1$ |
| Wild slopes: | [2] |
| Galois mean slope: | $20/11$ |
| Galois splitting model: | Not computed |