Defining polynomial
\(x^{9} + 3 x^{4} + 6 x^{3} + 3\) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $9$ |
Ramification exponent $e$: | $9$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $12$ |
Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
This field is not Galois over $\Q_{3}.$ | |
Visible slopes: | $[3/2, 3/2]$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 3 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{3}$ |
Relative Eisenstein polynomial: | \( x^{9} + 3 x^{4} + 6 x^{3} + 3 \) |
Ramification polygon
Residual polynomials: | $z^{4} + z + 2$ |
Associated inertia: | $4$ |
Indices of inseparability: | $[4, 3, 0]$ |