Defining polynomial
|
\(x^{10} + 38\)
|
Invariants
| Base field: | $\Q_{19}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $10$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $9$ |
| Discriminant root field: | $\Q_{19}(\sqrt{19})$ |
| Root number: | $-i$ |
| $\card{ \Aut(K/\Q_{ 19 }) }$: | $2$ |
| This field is not Galois over $\Q_{19}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{19}(\sqrt{19})$, 19.5.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{19}$ |
| Relative Eisenstein polynomial: |
\( x^{10} + 38 \)
|
Ramification polygon
| Residual polynomials: | $z^{9} + 10z^{8} + 7z^{7} + 6z^{6} + z^{5} + 5z^{4} + z^{3} + 6z^{2} + 7z + 10$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $D_{10}$ (as 10T3) |
| Inertia group: | $C_{10}$ (as 10T1) |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $10$ |
| Wild slopes: | None |
| Galois mean slope: | $9/10$ |
| Galois splitting model: | Not computed |