Defining polynomial
| \( x^{13} - x + 4 \) |
Invariants
| Base field: | $\Q_{19}$ |
| Degree $d$: | $13$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $13$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{19}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 19 })|$: | $13$ |
| 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: | 19.13.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{13} - x + 4 \) |
| Relative Eisenstein polynomial: | $ x - 19 \in\Q_{19}(t)[x]$ |