Defining polynomial
| \( x^{14} - 21 x^{12} - 21 x^{11} + 21 x^{10} + 21 x^{9} - 14 x^{7} + 7 x^{6} + 7 x^{5} + 7 x^{3} - 7 x^{2} - 14 \) |
Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$ : | $14$ |
| Ramification exponent $e$ : | $14$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $15$ |
| Discriminant root field: | $\Q_{7}(\sqrt{7})$ |
| Root number: | $i$ |
| $|\Aut(K/\Q_{ 7 })|$: | $2$ |
| This field is not Galois over $\Q_{7}$. | |
Intermediate fields
| $\Q_{7}(\sqrt{7})$, 7.7.7.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}$ |
| Relative Eisenstein polynomial: | \( x^{14} - 21 x^{12} - 21 x^{11} + 21 x^{10} + 21 x^{9} - 14 x^{7} + 7 x^{6} + 7 x^{5} + 7 x^{3} - 7 x^{2} - 14 \) |