Defining polynomial
|
\(x^{12} + x^{6} + x^{5} + x^{4} + x^{2} + 2\)
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $1$ |
| Residue field degree $f$: | $12$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{3})$ $=$$\Gal(K/\Q_{3})$: | $C_{12}$ |
| This field is Galois and abelian over $\Q_{3}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $531440 = (3^{ 12 } - 1)$ |
Intermediate fields
| $\Q_{3}(\sqrt{2})$, 3.3.1.0a1.1, 3.4.1.0a1.1, 3.6.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 3.12.1.0a1.1 $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{12} + x^{6} + x^{5} + x^{4} + x^{2} + 2 \)
|
| Relative Eisenstein polynomial: |
\( x - 3 \)
$\ \in\Q_{3}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.