Defining polynomial
\(x^{4} - 20\)  |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $4$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $6$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-5})$ |
| Root number: | $-i$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}.$ | |
Intermediate fields
| $\Q_{2}(\sqrt{5})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} - x + 1 \) |
| Relative Eisenstein polynomial: | \( x^{2} - 4 t - 6 \)$\ \in\Q_{2}(t)[x]$ |
Invariants of the Galois closure
| Galois group: | $D_4$ (as 4T3) |
| Inertia group: | Intransitive group isomorphic to $C_2^2$ |
| Unramified degree: | $2$ |
| Tame degree: | $1$ |
| Wild slopes: | [2, 3] |
| Galois mean slope: | $2$ |
| Galois splitting model: | $x^{4} - 20$ |