Defining polynomial
| \( x^{8} + 8 x^{2} + 52 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $8$ |
| Ramification exponent $e$ : | $8$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $16$ |
| Discriminant root field: | $\Q_{2}(\sqrt{*})$ |
| Root number: | $-1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $2$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| 2.4.6.7 |
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: | \( x^{8} - 8 x^{7} + 10 x^{6} - 344 x^{5} + 2814 x^{4} + 8384 x^{3} + 54080 x^{2} + 90204 x + 102662 \) |