Defining polynomial
| \( x^{7} - 19 \) |
Invariants
| Base field: | $\Q_{19}$ |
| Degree $d$ : | $7$ |
| Ramification exponent $e$ : | $7$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $6$ |
| Discriminant root field: | $\Q_{19}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 19 })|$: | $1$ |
| This field is not Galois 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: | $\Q_{19}$ |
| Relative Eisenstein polynomial: | \( x^{7} - 19 \) |
Invariants of the Galois closure
| Galois group: | $F_7$ (as 7T4) |
| Inertia group: | $C_7$ |
| Unramified degree: | $6$ |
| Tame degree: | $7$ |
| Wild slopes: | None |
| Galois mean slope: | $6/7$ |
| Galois splitting model: | $x^{7} - 19$ |