Defining polynomial
\(x^{3} + 17\) |
Invariants
Base field: | $\Q_{17}$ |
Degree $d$: | $3$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $2$ |
Discriminant root field: | $\Q_{17}(\sqrt{3})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 17 }) }$: | $1$ |
This field is not Galois over $\Q_{17}.$ | |
Visible slopes: | None |
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: | \( x^{3} + 17 \) |
Ramification polygon
Residual polynomials: | $z^{2} + 3z + 3$ |
Associated inertia: | $2$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $S_3$ (as 3T2) |
Inertia group: | $C_3$ (as 3T1) |
Wild inertia group: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $3$ |
Wild slopes: | None |
Galois mean slope: | $2/3$ |
Galois splitting model: | $x^{3} - 17$ |