Defining polynomial
|
\(x^{4} + x + 1\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $4$ |
| Ramification index $e$: | $1$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{2})$ $=$$\Gal(K/\Q_{2})$: | $C_4$ |
| This field is Galois and abelian over $\Q_{2}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | $[1]$ |
| Roots of unity: | $30 = (2^{ 4 } - 1) \cdot 2$ |
Intermediate fields
| $\Q_{2}(\sqrt{5})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 2.4.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{4} + x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x - 2 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.
Invariants of the Galois closure
| Galois degree: | $4$ |
| Galois group: | $C_4$ (as 4T1) |
| Inertia group: | trivial |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[]$ |
| Galois mean slope: | $0.0$ |
| Galois splitting model: | $x^{4} - x^{3} + x^{2} - x + 1$ |