Defining polynomial
|
\(x^{4} + 2 x + 2\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $4$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $4$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $1$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[4/3, 4/3]$ |
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: |
\( x^{4} + 2 x + 2 \)
|
Ramification polygon
| Residual polynomials: | $z + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 1, 0]$ |
Invariants of the Galois closure
| Galois group: | $S_4$ (as 4T5) |
| Inertia group: | $A_4$ (as 4T4) |
| Wild inertia group: | $C_2^2$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | $[4/3, 4/3]$ |
| Galois mean slope: | $7/6$ |
| Galois splitting model: |
$x^{4} + 2 x + 2$
|