Defining polynomial
|
\(x^{9} + 6 x + 3\)
|
Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $9$ |
| Ramification exponent $e$: | $9$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $9$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3\cdot 2})$ |
| Root number: | $-i$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[9/8, 9/8]$ |
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: |
\( x^{9} + 6 x + 3 \)
|
Ramification polygon
Data not computedInvariants of the Galois closure
| Galois group: | $F_9:C_2$ (as 9T19) |
| Inertia group: | $F_9$ (as 9T15) |
| Wild inertia group: | $C_3^2$ |
| Unramified degree: | $2$ |
| Tame degree: | $8$ |
| Wild slopes: | $[9/8, 9/8]$ |
| Galois mean slope: | $79/72$ |
| Galois splitting model: | $x^{9} - 3 x^{8} + 12 x^{5} + 12 x^{4} - 12 x + 4$ |