Defining polynomial
\(x^{10} + 2 x^{3} + 2\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $10$ |
Ramification exponent $e$: | $10$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $12$ |
Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $2$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[8/5]$ |
Intermediate fields
2.5.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{2}$ |
Relative Eisenstein polynomial: | \( x^{10} + 2 x^{3} + 2 \) |
Ramification polygon
Residual polynomials: | $z + 1$,$z^{8} + z^{6} + 1$ |
Associated inertia: | $1$,$4$ |
Indices of inseparability: | $[3, 0]$ |
Invariants of the Galois closure
Galois group: | $C_2^4:F_5$ (as 10T25) |
Inertia group: | $C_2^4:C_5$ (as 10T8) |
Wild inertia group: | $C_2^4$ |
Unramified degree: | $4$ |
Tame degree: | $5$ |
Wild slopes: | $[8/5, 8/5, 8/5, 8/5]$ |
Galois mean slope: | $31/20$ |
Galois splitting model: | $x^{10} - x^{8} + 10 x^{6} + 10 x^{4} + 5 x^{2} - 5$ |