Defining polynomial
\(x^{12} + 3 x^{2} + 3\) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $12$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $13$ |
Discriminant root field: | $\Q_{3}(\sqrt{3})$ |
Root number: | $-i$ |
$\card{ \Aut(K/\Q_{ 3 }) }$: | $2$ |
This field is not Galois over $\Q_{3}.$ | |
Visible slopes: | $[5/4]$ |
Intermediate fields
$\Q_{3}(\sqrt{3\cdot 2})$, 3.4.3.1, 3.6.6.6 |
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: | \( x^{12} + 3 x^{2} + 3 \) |
Ramification polygon
Residual polynomials: | $z^{2} + 1$,$z^{9} + z^{6} + 1$ |
Associated inertia: | $2$,$2$ |
Indices of inseparability: | $[2, 0]$ |