Defining polynomial
|
\(x^{5} + 10 x^{2} + 5\)
|
Invariants
| Base field: | $\Q_{5}$ |
|
| Degree $d$: | $5$ |
|
| Ramification index $e$: | $5$ |
|
| Residue field degree $f$: | $1$ |
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| Discriminant exponent $c$: | $6$ |
|
| 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: | $[\frac{3}{2}]$ | |
| Visible Swan slopes: | $[\frac{1}{2}]$ | |
| Means: | $\langle\frac{2}{5}\rangle$ | |
| Rams: | $(\frac{1}{2})$ | |
| 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: |
\( x^{5} + 10 x^{2} + 5 \)
|
Ramification polygon
| Residual polynomials: | $z^2 + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
| Galois degree: | $10$ |
| Galois group: | $D_5$ (as 5T2) |
| Inertia group: | $D_5$ (as 5T2) |
| Wild inertia group: | $C_5$ |
| Galois unramified degree: | $1$ |
| Galois tame degree: | $2$ |
| Galois Artin slopes: | $[\frac{3}{2}]$ |
| Galois Swan slopes: | $[\frac{1}{2}]$ |
| Galois mean slope: | $1.3$ |
| Galois splitting model: | $x^{5} - 5 x^{2} - 3$ |