Defining polynomial
\(x^{3} - x + 4\)  |
Invariants
| Base field: | $\Q_{19}$ |
| Degree $d$: | $3$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{19}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 19 })|$: | $3$ |
| 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: | 19.3.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{3} - x + 4 \) |
| Relative Eisenstein polynomial: | \( x - 19 \)$\ \in\Q_{19}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_3$ (as 3T1) |
| Inertia group: | trivial |
| Unramified degree: | $3$ |
| Tame degree: | $1$ |
| Wild slopes: | None |
| Galois mean slope: | $0$ |
| Galois splitting model: | $x^{3} - x^{2} - 2 x + 1$ |