Defining polynomial
|
\(x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 598 x^{5} + 750 x^{4} + 640 x^{3} + 280 x^{2} + 40 x + 17\)
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $5$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $8$ |
| Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $2$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{3}(\sqrt{2})$, 3.5.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{3}(\sqrt{2})$ $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{2} + 2 x + 2 \)
|
| Relative Eisenstein polynomial: |
\( x^{5} + 3 \)
$\ \in\Q_{3}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{4} + 2z^{3} + z^{2} + z + 2$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $F_5$ (as 10T4) |
| Inertia group: | Intransitive group isomorphic to $C_5$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $4$ |
| Tame degree: | $5$ |
| Wild slopes: | None |
| Galois mean slope: | $4/5$ |
| Galois splitting model: | $x^{10} - 5 x^{9} + 5 x^{8} + 10 x^{7} - 15 x^{6} - 8 x^{5} + 40 x^{3} - 35 x^{2} + 10 x - 4$ |