Defining polynomial
\( x^{9} + 3 x^{6} + 3 x^{5} + 3 x^{3} + 6 \) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $9$ |
Ramification exponent $e$: | $9$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $13$ |
Discriminant root field: | $\Q_{3}(\sqrt{3\cdot 2})$ |
Root number: | $-i$ |
$|\Aut(K/\Q_{ 3 })|$: | $3$ |
This field is not Galois over $\Q_{3}.$ |
Intermediate fields
3.3.3.2 |
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: | \( x^{9} + 3 x^{6} + 3 x^{5} + 3 x^{3} + 6 \) |