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: | $x + 3$ |