Defining polynomial
|
\(x^{19} + 2\)
|
Invariants
| Base field: | $\Q_{2}$ |
|
| Degree $d$: | $19$ |
|
| Ramification index $e$: | $19$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $18$ |
|
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{2})$: | $C_1$ | |
| This field is not Galois over $\Q_{2}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | $[19]$ | |
| Roots of unity: | $2$ |
|
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 2 }$. |
Canonical tower
| Unramified subfield: | $\Q_{2}$ |
|
| Relative Eisenstein polynomial: |
\( x^{19} + 2 \)
|
Ramification polygon
| Residual polynomials: | $z^{18} + z^{17} + z^{16} + z^{15} + z^2 + z + 1$ |
| Associated inertia: | $18$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $342$ |
| Galois group: | $F_{19}$ (as 19T6) |
| Inertia group: | $C_{19}$ (as 19T1) |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $18$ |
| Galois tame degree: | $19$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.9473684210526315$ |
| Galois splitting model: | $x^{19} - 2$ |