Defining polynomial
|
$( x^{2} + x + 1 )^{5} + 2$
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $10$ |
| Ramification index $e$: | $5$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $8$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$: | $C_2$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | $[5]$ |
| Roots of unity: | $6 = (2^{ 2 } - 1) \cdot 2$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$, 2.1.5.4a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{2} + x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{5} + 2 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{4} + z^{3} + 1$ |
| 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: | $x^{10} - 5 x^{9} + 15 x^{8} - 30 x^{7} + 45 x^{6} - 49 x^{5} + 35 x^{4} - 10 x^{3} - 5 x^{2} + 5 x - 1$ |