Defining polynomial
\(x^{15} + 2\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $15$ |
Ramification exponent $e$: | $15$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $14$ |
Discriminant root field: | $\Q_{2}$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $1$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | None |
Intermediate fields
2.3.2.1, 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: | \( x^{15} + 2 \) |
Ramification polygon
Residual polynomials: | $z^{14} + z^{13} + z^{12} + z^{11} + z^{10} + z^{9} + z^{8} + z^{7} + z^{6} + z^{5} + z^{4} + z^{3} + z^{2} + z + 1$ |
Associated inertia: | $4$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_{15}:C_4$ (as 15T6) |
Inertia group: | $C_{15}$ (as 15T1) |
Wild inertia group: | $C_1$ |
Unramified degree: | $4$ |
Tame degree: | $15$ |
Wild slopes: | None |
Galois mean slope: | $14/15$ |
Galois splitting model: | Not computed |