Defining polynomial
| \( x^{13} - 2 x + 3 \) |
Invariants
| Base field: | $\Q_{17}$ |
| Degree $d$: | $13$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $13$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{17}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 17 })|$: | $13$ |
| This field is Galois and abelian over $\Q_{17}.$ | |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 17 }$. |
Unramified/totally ramified tower
| Unramified subfield: | 17.13.0.1 $\cong \Q_{17}(t)$ where $t$ is a root of \( x^{13} - 2 x + 3 \) |
| Relative Eisenstein polynomial: | $ x - 17 \in\Q_{17}(t)[x]$ |