Defining polynomial
\(x^{13} + 2\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $13$ |
Ramification exponent $e$: | $13$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $12$ |
Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $1$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | None |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 2 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{2}$ |
Relative Eisenstein polynomial: | \( x^{13} + 2 \) |
Ramification polygon
Residual polynomials: | $z^{12} + z^{11} + z^{8} + z^{7} + z^{4} + z^{3} + 1$ |
Associated inertia: | $12$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $F_{13}$ (as 13T6) |
Inertia group: | $C_{13}$ (as 13T1) |
Wild inertia group: | $C_1$ |
Unramified degree: | $12$ |
Tame degree: | $13$ |
Wild slopes: | None |
Galois mean slope: | $12/13$ |
Galois splitting model: | $x^{13} - 13 x^{12} + 78 x^{11} - 286 x^{10} + 715 x^{9} - 1287 x^{8} + 1716 x^{7} - 1716 x^{6} + 1287 x^{5} - 715 x^{4} + 286 x^{3} - 78 x^{2} + 13 x - 3$ |