Defining polynomial
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\(x^{2} + 2 x + 6\)
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Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $2$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $2$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-5})$ |
| Root number: | $i$ |
| $\card{ \Gal(K/\Q_{ 2 }) }$: | $2$ |
| This field is Galois and abelian over $\Q_{2}.$ | |
| Visible slopes: | $[2]$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 2 }$. |
Unramified/totally ramified 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 group: | $C_2$ (as 2T1) |
| Inertia group: | $C_2$ (as 2T1) |
| Wild inertia group: | $C_2$ |
| Unramified degree: | $1$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2]$ |
| Galois mean slope: | $1$ |
| Galois splitting model: |
$x^{2} + 2 x - 2$
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