Defining polynomial
|
\(x^{5} + 10 x^{4} + 5\)
|
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $5$ |
| Ramification index $e$: | $5$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $8$ |
| Discriminant root field: | $\Q_{5}(\sqrt{2})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{5})$: | $C_1$ |
| This field is not Galois over $\Q_{5}.$ | |
| Visible Artin slopes: | $[2]$ |
| Visible Swan slopes: | $[1]$ |
| Means: | $\langle\frac{4}{5}\rangle$ |
| Rams: | $(1)$ |
| 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: |
\( x^{5} + 10 x^{4} + 5 \)
|
Ramification polygon
| Residual polynomials: | $z^{4} + 2$ |
| Associated inertia: | $4$ |
| Indices of inseparability: | $[4, 0]$ |
Invariants of the Galois closure
| Galois degree: | $20$ |
| Galois group: | $F_5$ (as 5T3) |
| Inertia group: | $C_5$ (as 5T1) |
| Wild inertia group: | $C_5$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2]$ |
| Galois Swan slopes: | $[1]$ |
| Galois mean slope: | $1.6$ |
| Galois splitting model: | $x^{5} - 10 x^{3} - 180 x^{2} + 935 x - 1156$ |