Defining polynomial
|
\(x^{2} + 2 x + 6\)
|
Invariants
| Base field: | $\Q_{2}$ |
|
| Degree $d$: | $2$ |
|
| Ramification index $e$: | $2$ |
|
| Residue field degree $f$: | $1$ |
|
| Discriminant exponent $c$: | $2$ |
|
| Discriminant root field: | $\Q_{2}(\sqrt{-5})$ | |
| Root number: | $i$ | |
| $\Aut(K/\Q_{2})$ $=$ $\Gal(K/\Q_{2})$: | $C_2$ | |
| This field is Galois and abelian over $\Q_{2}.$ | ||
| Visible Artin slopes: | $[2]$ | |
| Visible Swan slopes: | $[1]$ | |
| Means: | $\langle\frac{1}{2}\rangle$ | |
| Rams: | $(1)$ | |
| Jump set: | $[1, 4]$ | |
| 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: |
\( x^{2} + 2 x + 6 \)
|
Ramification polygon
| Residual polynomials: | $z + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois degree: | $2$ |
| Galois group: | $C_2$ (as 2T1) |
| Inertia group: | $C_2$ (as 2T1) |
| Wild inertia group: | $C_2$ |
| Galois unramified degree: | $1$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2]$ |
| Galois Swan slopes: | $[1]$ |
| Galois mean slope: | $1.0$ |
| Galois splitting model: | $x^{2} - 3$ |