Defining polynomial
|
\(x^{4} + 2 x^{3} + 2 x^{2} + 2\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $4$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $6$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $-1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $1$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | $[2, 2]$ |
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: |
\( x^{4} + 2 x^{3} + 2 x^{2} + 2 \)
|
Ramification polygon
Data not computedInvariants of the Galois closure
| Galois group: | $A_4$ (as 4T4) |
| Inertia group: | $C_2^2$ (as 4T2) |
| Wild inertia group: | $C_2^2$ |
| Unramified degree: | $3$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2, 2]$ |
| Galois mean slope: | $3/2$ |
| Galois splitting model: |
$x^{4} + 2 x^{3} + 2 x^{2} + 2$
|
Additional information
This is the only degree $4$ extension of $\Q_p$, for any $p$, which has Galois group $A_4$.