Defining polynomial
\(x^{10} - 2 x^{5} - 2\) ![]() |
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
$\Q_{2}(\sqrt{-5})$, 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: | \( x^{10} - 160 x^{9} - 40460 x^{8} - 2808640 x^{7} - 98032360 x^{6} - 998918222 x^{5} - 8072348440 x^{4} + 192345129720 x^{3} - 1170688441400 x^{2} + 3109663379160 x - 3510500485802 \) ![]() |