Defining polynomial
|
\(x^{9} + 3 x^{6} + 6 x^{5} + 6 x^{3} + 3\)
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $9$ |
| Ramification exponent $e$: | $9$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $13$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3})$ |
| Root number: | $i$ |
| $\card{ \Aut(K/\Q_{ 3 }) }$: | $3$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible slopes: | $[3/2, 5/3]$ |
Intermediate fields
| 3.3.3.1 |
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: |
\( x^{9} + 3 x^{6} + 6 x^{5} + 6 x^{3} + 3 \)
|
Ramification polygon
| Residual polynomials: | $z^{2} + 2$,$z^{3} + 1$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[5, 3, 0]$ |