Defining polynomial
|
\(x^{11} + 5\)
|
Invariants
| Base field: | $\Q_{5}$ |
|
| Degree $d$: | $11$ |
|
| Ramification index $e$: | $11$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $10$ |
|
| Discriminant root field: | $\Q_{5}$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{5})$: | $C_1$ | |
| This field is not Galois over $\Q_{5}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $4 = (5 - 1)$ |
|
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 5 }$. |
Canonical tower
| Unramified subfield: | $\Q_{5}$ |
|
| Relative Eisenstein polynomial: |
\( x^{11} + 5 \)
|
Ramification polygon
| Residual polynomials: | $z^{10} + z^9 + 2 z^5 + 2 z^4 + 1$ |
| Associated inertia: | $5$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $55$ |
| Galois group: | $C_{11}:C_5$ (as 11T3) |
| Inertia group: | $C_{11}$ (as 11T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $5$ |
| Galois tame degree: | $11$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.9090909090909091$ |
| Galois splitting model: | $x^{11} - 55 x^{9} + 1100 x^{7} - 9625 x^{5} + 34375 x^{3} - 34375 x - 12675$ |