Defining polynomial
| \( x^{7} - 2 x + 9 \) |
Invariants
| Base field: | $\Q_{43}$ |
| Degree $d$ : | $7$ |
| Ramification exponent $e$ : | $1$ |
| Residue field degree $f$ : | $7$ |
| Discriminant exponent $c$ : | $0$ |
| Discriminant root field: | $\Q_{43}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 43 })|$: | $7$ |
| This field is Galois and abelian over $\Q_{43}$. | |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 43 }$. |
Unramified/totally ramified tower
| Unramified subfield: | 43.7.0.1 $\cong \Q_{43}(t)$ where $t$ is a root of \( x^{7} - 2 x + 9 \) |
| Relative Eisenstein polynomial: | $ x - 43 \in\Q_{43}(t)[x]$ |