Defining polynomial
\(x^{6} + 6 x^{2} + 6\)
|
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $6$ |
Ramification exponent $e$: | $6$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $7$ |
Discriminant root field: | $\Q_{3}(\sqrt{3})$ |
Root number: | $i$ |
$\card{ \Gal(K/\Q_{ 3 }) }$: | $6$ |
This field is Galois over $\Q_{3}.$ | |
Visible slopes: | $[3/2]$ |
Intermediate fields
$\Q_{3}(\sqrt{3})$, 3.3.3.1 x3 |
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: |
\( x^{6} + 6 x^{2} + 6 \)
|
Ramification polygon
Residual polynomials: | $2z^{2} + 1$,$z^{3} + 2$ |
Associated inertia: | $1$,$1$ |
Indices of inseparability: | $[2, 0]$ |