Defining polynomial
\(x^{8} + 801\) |
Invariants
Base field: | $\Q_{89}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $8$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $7$ |
Discriminant root field: | $\Q_{89}(\sqrt{89})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 89 }) }$: | $8$ |
This field is Galois and abelian over $\Q_{89}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{89}(\sqrt{89})$, 89.4.3.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{89}$ |
Relative Eisenstein polynomial: | \( x^{8} + 801 \) |
Ramification polygon
Residual polynomials: | $z^{7} + 8z^{6} + 28z^{5} + 56z^{4} + 70z^{3} + 56z^{2} + 28z + 8$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_8$ (as 8T1) |
Inertia group: | $C_8$ (as 8T1) |
Wild inertia group: | $C_1$ |
Unramified degree: | $1$ |
Tame degree: | $8$ |
Wild slopes: | None |
Galois mean slope: | $7/8$ |
Galois splitting model: | Not computed |