Defining polynomial
\(x^{8} + 21\) |
Invariants
Base field: | $\Q_{7}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $8$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $7$ |
Discriminant root field: | $\Q_{7}(\sqrt{7\cdot 3})$ |
Root number: | $-i$ |
$\card{ \Aut(K/\Q_{ 7 }) }$: | $2$ |
This field is not Galois over $\Q_{7}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{7}(\sqrt{7})$, 7.4.3.2 |
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: | \( x^{8} + 21 \) |
Ramification polygon
Residual polynomials: | $z^{7} + z^{6} + 1$ |
Associated inertia: | $2$ |
Indices of inseparability: | $[0]$ |