Defining polynomial
\(x^{10} + 2 x^{8} - 2 x^{7} + 2 x^{5} + 2 x^{2} + 6\) ![]() |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $10$ |
Ramification exponent $e$: | $10$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $14$ |
Discriminant root field: | $\Q_{2}(\sqrt{-5})$ |
Root number: | $i$ |
$|\Aut(K/\Q_{ 2 })|$: | $2$ |
This field is not Galois over $\Q_{2}.$ |
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: | \( x^{10} + 2 x^{8} - 2 x^{7} + 2 x^{5} + 2 x^{2} + 6 \) ![]() |
Invariants of the Galois closure
Galois group: | $((C_2^4 : C_5):C_4)\times C_2$ (as 10T29) |
Inertia group: | $C_2 \times (C_2^4 : C_5)$ |
Unramified degree: | $4$ |
Tame degree: | $5$ |
Wild slopes: | [8/5, 8/5, 8/5, 8/5, 2] |
Galois mean slope: | $71/40$ |
Galois splitting model: | $x^{10} + 5 x^{8} - 6 x^{7} + 4 x^{6} - 4 x^{5} + 30 x^{4} - 42 x^{3} + 39 x^{2} - 36 x + 23$ |