Defining polynomial
\(x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57\) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $9$ |
Ramification exponent $e$: | $9$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $22$ |
Discriminant root field: | $\Q_{3}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 3 }) }$: | $9$ |
This field is Galois and abelian over $\Q_{3}.$ | |
Visible slopes: | $[2, 3]$ |
Intermediate fields
3.3.4.2 |
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: | \( x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57 \) |
Ramification polygon
Residual polynomials: | $2z^{2} + 1$,$z^{6} + 2$ |
Associated inertia: | $1$,$1$ |
Indices of inseparability: | $[14, 6, 0]$ |