Defining polynomial
|
\(x^{5} + 41\)
|
Invariants
| Base field: | $\Q_{41}$ |
| Degree $d$: | $5$ |
| Ramification index $e$: | $5$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $4$ |
| Discriminant root field: | $\Q_{41}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{41})$ $=$$\Gal(K/\Q_{41})$: | $C_5$ |
| This field is Galois and abelian over $\Q_{41}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $40 = (41 - 1)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 41 }$. |
Canonical tower
| Unramified subfield: | $\Q_{41}$ |
| Relative Eisenstein polynomial: |
\( x^{5} + 41 \)
|
Ramification polygon
| Residual polynomials: | $z^4 + 5 z^3 + 10 z^2 + 10 z + 5$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $5$ |
| Galois group: | $C_5$ (as 5T1) |
| Inertia group: | $C_5$ (as 5T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $1$ |
| Galois tame degree: | $5$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.8$ |
| Galois splitting model: | $x^{5} - x^{4} - 16 x^{3} - 5 x^{2} + 21 x + 9$ |