Defining polynomial
| \( x^{9} + 9 x^{8} + 21 x^{6} + 18 x^{5} + 9 x^{3} + 24 \) |
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: | $1$ |
| $|\Gal(K/\Q_{ 3 })|$: | $9$ |
| This field is Galois and abelian over $\Q_{3}$. | |
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: | \( x^{9} + 9 x^{8} + 21 x^{6} + 18 x^{5} + 9 x^{3} + 24 \) |