Defining polynomial
| \( x^{14} + 4 x^{11} - 3 x^{10} + 4 x^{9} + 2 x^{8} + 2 x^{7} - 3 x^{6} + 2 x^{5} - 2 x^{4} - 2 x^{3} - x^{2} - 2 x + 1 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $14$ |
| Ramification exponent $e$ : | $2$ |
| Residue field degree $f$ : | $7$ |
| Discriminant exponent $c$ : | $21$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-2})$ |
| Root number: | $-i$ |
| $|\Gal(K/\Q_{ 2 })|$: | $14$ |
| This field is Galois and abelian over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{-2})$, 2.7.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 2.7.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{7} - x + 1 \) |
| Relative Eisenstein polynomial: | $ x^{2} + \left(12 t^{6} + 8 t^{5} + 4 t^{4} + 12 t^{2} + 4\right) x + 4 t^{6} + 12 t^{5} + 8 t^{4} + 14 t^{3} + 12 t^{2} + 8 t \in\Q_{2}(t)[x]$ |