Defining polynomial
\(x^{7} + 11\) |
Invariants
Base field: | $\Q_{11}$ |
Degree $d$: | $7$ |
Ramification exponent $e$: | $7$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{11}$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 11 }) }$: | $1$ |
This field is not Galois over $\Q_{11}.$ | |
Visible slopes: | None |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 11 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{11}$ |
Relative Eisenstein polynomial: | \( x^{7} + 11 \) |
Ramification polygon
Residual polynomials: | $z^{6} + 7z^{5} + 10z^{4} + 2z^{3} + 2z^{2} + 10z + 7$ |
Associated inertia: | $3$ |
Indices of inseparability: | $[0]$ |