Defining polynomial
|
\(x^{10} + 15 x^{4} + 5\)
|
Invariants
| Base field: | $\Q_{5}$ |
|
| Degree $d$: | $10$ |
|
| Ramification index $e$: | $10$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $13$ |
|
| Discriminant root field: | $\Q_{5}(\sqrt{5})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{5})$ $=$ $\Gal(K/\Q_{5})$: | $D_5$ | |
| This field is Galois over $\Q_{5}.$ | ||
| Visible Artin slopes: | $[\frac{3}{2}]$ | |
| Visible Swan slopes: | $[\frac{1}{2}]$ | |
| Means: | $\langle\frac{2}{5}\rangle$ | |
| Rams: | $(1)$ | |
| Jump set: | undefined | |
| Roots of unity: | $4 = (5 - 1)$ |
|
Intermediate fields
| $\Q_{5}(\sqrt{5})$, 5.1.5.6a1.2 x5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{5}$ |
|
| Relative Eisenstein polynomial: |
\( x^{10} + 15 x^{4} + 5 \)
|
Ramification polygon
| Residual polynomials: | $z^5 + 2$,$2 z^4 + 3$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[4, 0]$ |
Invariants of the Galois closure
| Galois degree: | $10$ |
| Galois group: | $D_5$ (as 10T2) |
| Inertia group: | $D_5$ (as 10T2) |
| 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: | $x^{10} + 5 x^{8} - 10 x^{7} + 5 x^{6} - 28 x^{5} + 15 x^{4} - 10 x^{3} + 30 x^{2} + 40 x + 16$ |