Defining polynomial
|
\(x^{17} + 3\)
|
Invariants
| Base field: | $\Q_{3}$ |
|
| Degree $d$: | $17$ |
|
| Ramification index $e$: | $17$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $16$ |
|
| Discriminant root field: | $\Q_{3}(\sqrt{2})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{3})$: | $C_1$ | |
| This field is not Galois over $\Q_{3}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $2 = (3 - 1)$ |
|
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 3 }$. |
Canonical tower
| Unramified subfield: | $\Q_{3}$ |
|
| Relative Eisenstein polynomial: |
\( x^{17} + 3 \)
|
Ramification polygon
| Residual polynomials: | $z^{16} + 2 z^{15} + z^{14} + 2 z^{13} + z^{12} + 2 z^{11} + z^{10} + 2 z^9 + z^8 + z^7 + 2 z^6 + z^5 + 2 z^4 + z^3 + 2 z^2 + z + 2$ |
| Associated inertia: | $16$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $272$ |
| Galois group: | $F_{17}$ (as 17T5) |
| Inertia group: | $C_{17}$ (as 17T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $16$ |
| Galois tame degree: | $17$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.9411764705882353$ |
| Galois splitting model: | $x^{17} - 3$ |