Defining polynomial
| \( x^{4} + 4 x + 2 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $4$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $8$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $1$ |
| This field is not Galois over $\Q_{2}.$ | |
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: | \( x^{4} + 4 x + 2 \) |
Invariants of the Galois closure
| Galois group: | $S_4$ (as 4T5) |
| Inertia group: | $A_4$ |
| Unramified degree: | $2$ |
| Tame degree: | $3$ |
| Wild slopes: | [8/3, 8/3] |
| Galois mean slope: | $13/6$ |
| Galois splitting model: | $x^{4} + 4 x + 2$ |