Defining polynomial
|
\(x^{5} + 19\)
|
Invariants
| Base field: | $\Q_{19}$ |
| Degree $d$: | $5$ |
| Ramification exponent $e$: | $5$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $4$ |
| Discriminant root field: | $\Q_{19}$ |
| 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: |
\( x^{5} + 19 \)
|
Ramification polygon
| Residual polynomials: | $z^{4} + 5z^{3} + 10z^{2} + 10z + 5$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |