Defining polynomial
|
\(x^{6} + 3\)
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $6$ |
| Ramification exponent $e$: | $6$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $11$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3\cdot 2})$ |
| Root number: | $-i$ |
| $\card{ \Gal(K/\Q_{ 3 }) }$: | $6$ |
| This field is Galois over $\Q_{3}.$ | |
| Visible slopes: | $[5/2]$ |
Intermediate fields
| $\Q_{3}(\sqrt{3\cdot 2})$, 3.3.5.1 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{3}$ |
| Relative Eisenstein polynomial: |
\( x^{6} + 3 \)
|
Ramification polygon
| Residual polynomials: | $2z^{2} + 1$,$z^{3} + 2$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[6, 0]$ |
Invariants of the Galois closure
| Galois group: | $S_3$ (as 6T2) |
| Inertia group: | $S_3$ (as 6T2) |
| Wild inertia group: | $C_3$ |
| Unramified degree: | $1$ |
| Tame degree: | $2$ |
| Wild slopes: | $[5/2]$ |
| Galois mean slope: | $11/6$ |
| Galois splitting model: | $x^{6} + 3$ |