Defining polynomial
\(x^{5} - x + 5\)  |
Invariants
| Base field: | $\Q_{19}$ |
| Degree $d$: | $5$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{19}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 19 })|$: | $5$ |
| 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: | 19.5.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{5} - x + 5 \) |
| Relative Eisenstein polynomial: | \( x - 19 \)$\ \in\Q_{19}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $C_5$ (as 5T1) |
| Inertia group: | trivial |
| Unramified degree: | $5$ |
| Tame degree: | $1$ |
| Wild slopes: | None |
| Galois mean slope: | $0$ |
| Galois splitting model: | $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ |