Defining polynomial
| \( x^{11} + x^{2} - x + 1 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $11$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $11$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 3 })|$: | $11$ |
| This field is Galois and abelian over $\Q_{3}.$ | |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 3 }$. |
Unramified/totally ramified tower
| Unramified subfield: | 3.11.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{11} + x^{2} - x + 1 \) |
| Relative Eisenstein polynomial: | $ x - 3 \in\Q_{3}(t)[x]$ |