Defining polynomial
|
\(x^{7} + 3\)
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $7$ |
| Ramification exponent $e$: | $7$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $6$ |
| Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | None |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 3 }$. |
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{3}$ |
| Relative Eisenstein polynomial: |
\( x^{7} + 3 \)
|
Ramification polygon
| Residual polynomials: | $z^{6} + z^{5} + 2z^{3} + 2z^{2} + 1$ |
| Associated inertia: | $6$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $F_7$ (as 7T4) |
| Inertia group: | $C_7$ (as 7T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $6$ |
| Tame degree: | $7$ |
| Wild slopes: | None |
| Galois mean slope: | $6/7$ |
| Galois splitting model: | $x^{7} - 3$ |