Defining polynomial
| \( x^{8} + 12 x^{2} + 4 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $8$ |
| Ramification exponent $e$ : | $8$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $12$ |
| Discriminant root field: | $\Q_{2}$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| $\Q_{2}(\sqrt{-*})$, 2.4.4.5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: | \( x^{8} - 8 x^{7} + 36 x^{6} - 254 x^{5} + 1814 x^{4} - 3772 x^{3} + 15366 x^{2} - 20980 x + 13726 \) |