Defining polynomial
\(x^{8} + 2 x^{7} + 2\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $8$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $14$ |
Discriminant root field: | $\Q_{2}$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[2, 2, 2]$ |
Intermediate fields
$\Q_{2}(\sqrt{-1})$, 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: | \( x^{8} + 2 x^{7} + 2 \) |
Ramification polygon
Residual polynomials: | $z^{7} + 1$ |
Associated inertia: | $3$ |
Indices of inseparability: | $[7, 7, 7, 0]$ |