Defining polynomial
| \( x^{14} + 21 x^{13} + 14 x^{12} + 7 x^{11} - 14 x^{10} - 7 x^{9} - 7 x^{8} - 7 x^{7} - 7 x^{6} - 21 x^{5} - 7 x^{4} - 14 x^{3} + 7 x^{2} + 21 \) |
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
| $\Q_{7}(\sqrt{7})$, 7.7.7.5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{7}$ |
| Relative Eisenstein polynomial: | \( x^{14} + 21 x^{13} + 14 x^{12} + 7 x^{11} - 14 x^{10} - 7 x^{9} - 7 x^{8} - 7 x^{7} - 7 x^{6} - 21 x^{5} - 7 x^{4} - 14 x^{3} + 7 x^{2} + 21 \) |