Defining polynomial
\(x^{11} + 3\) |
Invariants
Base field: | $\Q_{3}$ |
Degree $d$: | $11$ |
Ramification exponent $e$: | $11$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $10$ |
Discriminant root field: | $\Q_{3}$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 3 }) }$: | $1$ |
This field is not Galois over $\Q_{3}.$ | |
Visible slopes: | None |
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: | \( x^{11} + 3 \) |
Ramification polygon
Residual polynomials: | $z^{10} + 2z^{9} + z^{8} + z + 2$ |
Associated inertia: | $5$ |
Indices of inseparability: | $[0]$ |