Defining polynomial
|
\(x^{17} + 103\)
|
Invariants
| Base field: | $\Q_{103}$ |
| Degree $d$: | $17$ |
| Ramification index $e$: | $17$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $16$ |
| Discriminant root field: | $\Q_{103}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{103})$ $=$$\Gal(K/\Q_{103})$: | $C_{17}$ |
| This field is Galois and abelian over $\Q_{103}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $102 = (103 - 1)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 103 }$. |
Canonical tower
| Unramified subfield: | $\Q_{103}$ |
| Relative Eisenstein polynomial: |
\( x^{17} + 103 \)
|
Ramification polygon
| Residual polynomials: | $z^{16} + 17 z^{15} + 33 z^{14} + 62 z^{13} + 11 z^{12} + 8 z^{11} + 16 z^{10} + 84 z^9 + 2 z^8 + 2 z^7 + 84 z^6 + 16 z^5 + 8 z^4 + 11 z^3 + 62 z^2 + 33 z + 17$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $17$ |
| Galois group: | $C_{17}$ (as 17T1) |
| Inertia group: | $C_{17}$ (as 17T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $1$ |
| Galois tame degree: | $17$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.9411764705882353$ |
| Galois splitting model: | not computed |