Defining polynomial
|
\(x^{5} + 13\)
|
Invariants
| Base field: | $\Q_{13}$ |
| Degree $d$: | $5$ |
| Ramification index $e$: | $5$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $4$ |
| Discriminant root field: | $\Q_{13}(\sqrt{2})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{13})$: | $C_1$ |
| This field is not Galois over $\Q_{13}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $12 = (13 - 1)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 13 }$. |
Canonical tower
| Unramified subfield: | $\Q_{13}$ |
| Relative Eisenstein polynomial: |
\( x^{5} + 13 \)
|
Ramification polygon
| Residual polynomials: | $z^{4} + 5 z^{3} + 10 z^{2} + 10 z + 5$ |
| Associated inertia: | $4$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $20$ |
| Galois group: | $F_5$ (as 5T3) |
| Inertia group: | $C_5$ (as 5T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $5$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[]$ |
| Galois mean slope: | $0.8$ |
| Galois splitting model: | $x^{5} - 13$ |