Defining polynomial
\(x + 3\)  |
Invariants
| Base field: | $\Q_{17}$ |
| Degree $d$: | $1$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{17}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 17 })|$: | $1$ |
| 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: | $\Q_{17}$ |
| Relative Eisenstein polynomial: | \( x - 17 \) |
Invariants of the Galois closure
| Galois group: | $C_1$ (as 1T1) |
| Inertia group: | Trivial |
| Unramified degree: | $1$ |
| Tame degree: | $1$ |
| Wild slopes: | None |
| Galois mean slope: | $0$ |
| Galois splitting model: | $x + 3$ |