Defining polynomial
| \( x^{7} - 8 x + 4 \) |
Invariants
| Base field: | $\Q_{19}$ |
| Degree $d$ : | $7$ |
| Ramification exponent $e$ : | $1$ |
| Residue field degree $f$ : | $7$ |
| Discriminant exponent $c$ : | $0$ |
| Discriminant root field: | $\Q_{19}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 19 })|$: | $7$ |
| This field is Galois and abelian over $\Q_{19}$. | |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 19 }$. |
Unramified/totally ramified tower
| Unramified subfield: | 19.7.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{7} - 8 x + 4 \) |
| Relative Eisenstein polynomial: | $ x - 19 \in\Q_{19}(t)[x]$ |