Defining polynomial
| \( x^{5} + x^{2} - x + 5 \) |
Invariants
| Base field: | $\Q_{11}$ |
| Degree $d$: | $5$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{11}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 11 })|$: | $5$ |
| This field is Galois and abelian 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: | 11.5.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{5} + x^{2} - x + 5 \) |
| Relative Eisenstein polynomial: | $ x - 11 \in\Q_{11}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_5$ (as 5T1) |
| Inertia group: | trivial |
| Unramified degree: | $5$ |
| Tame degree: | $1$ |
| Wild slopes: | None |
| Galois mean slope: | $0$ |
| Galois splitting model: | $x^{5} - 10 x^{3} - 5 x^{2} + 10 x - 1$ |