Defining polynomial
\( x^{9} + 3 x^{8} + 45 x^{3} + 27 \) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $9$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $3$ |
Discriminant exponent $c$: | $12$ |
Discriminant root field: | $\Q_{3}$ |
Root number: | $1$ |
$|\Aut(K/\Q_{ 3 })|$: | $3$ |
This field is not Galois over $\Q_{3}.$ |
Intermediate fields
3.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 3.3.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{3} - x + 1 \) |
Relative Eisenstein polynomial: | $ x^{3} + \left(6 t^{2} + 6 t + 6\right) x^{2} + 6 t + 3 \in\Q_{3}(t)[x]$ |
Invariants of the Galois closure
Galois group: | $C_3\wr C_3$ (as 9T17) |
Inertia group: | Intransitive group isomorphic to $C_3^3$ |
Unramified degree: | $3$ |
Tame degree: | $1$ |
Wild slopes: | [2, 2, 2] |
Galois mean slope: | $52/27$ |
Galois splitting model: | $x^{9} - 597 x^{7} - 3184 x^{6} + 100296 x^{5} + 937887 x^{4} - 2237556 x^{3} - 42147603 x^{2} - 65335680 x + 188799061$ |