Defining polynomial
| \( x^{7} + x^{2} - x + 1 \) |
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$ : | $7$ |
| Ramification exponent $e$ : | $1$ |
| Residue field degree $f$ : | $7$ |
| Discriminant exponent $c$ : | $0$ |
| Discriminant root field: | $\Q_{3}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 3 })|$: | $7$ |
| 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: | 3.7.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{7} + x^{2} - x + 1 \) |
| Relative Eisenstein polynomial: | $ x - 3 \in\Q_{3}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_7$ (as 7T1) |
| Inertia group: | Trivial |
| Unramified degree: | $7$ |
| Tame degree: | $1$ |
| Wild slopes: | None |
| Galois mean slope: | $0$ |
| Galois splitting model: | $x^{7} + x^{6} - 12 x^{5} - 7 x^{4} + 28 x^{3} + 14 x^{2} - 9 x + 1$ |