Defining polynomial
\(x^{11} + 19\) |
Invariants
Base field: | $\Q_{19}$ |
Degree $d$: | $11$ |
Ramification exponent $e$: | $11$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $10$ |
Discriminant root field: | $\Q_{19}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 19 }) }$: | $1$ |
This field is not Galois over $\Q_{19}.$ | |
Visible slopes: | None |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 19 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{19}$ |
Relative Eisenstein polynomial: | \( x^{11} + 19 \) |
Ramification polygon
Residual polynomials: | $z^{10} + 11z^{9} + 17z^{8} + 13z^{7} + 7z^{6} + 6z^{5} + 6z^{4} + 7z^{3} + 13z^{2} + 17z + 11$ |
Associated inertia: | $10$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $F_{11}$ (as 11T4) |
Inertia group: | $C_{11}$ (as 11T1) |
Wild inertia group: | $C_1$ |
Unramified degree: | $10$ |
Tame degree: | $11$ |
Wild slopes: | None |
Galois mean slope: | $10/11$ |
Galois splitting model: | $x^{11} - 19$ |