Defining polynomial
|
$( x^{2} + 2 x + 2 )^{5} + 3$
|
Invariants
| Base field: | $\Q_{3}$ |
|
| Degree $d$: | $10$ |
|
| Ramification index $e$: | $5$ |
|
| Residue field degree $f$: | $2$ |
|
| Discriminant exponent $c$: | $8$ |
|
| Discriminant root field: | $\Q_{3}(\sqrt{2})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{3})$: | $C_2$ | |
| This field is not Galois over $\Q_{3}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $8 = (3^{ 2 } - 1)$ |
|
Intermediate fields
| $\Q_{3}(\sqrt{2})$, 3.1.5.4a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical 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: | $z^4 + 2 z^3 + z^2 + z + 2$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $20$ |
| Galois group: | $F_5$ (as 10T4) |
| Inertia group: | Intransitive group isomorphic to $C_5$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $5$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.8$ |
| Galois splitting model: | $x^{10} - 5 x^{9} + 5 x^{8} + 10 x^{7} - 15 x^{6} - 8 x^{5} + 40 x^{3} - 35 x^{2} + 10 x - 4$ |