Defining polynomial
\(x^{8} + 2 x^{3} + 2\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $8$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $10$ |
Discriminant root field: | $\Q_{2}$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $1$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | $[10/7, 10/7, 10/7]$ |
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: | \( x^{8} + 2 x^{3} + 2 \) |
Ramification polygon
Residual polynomials: | $z + 1$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[3, 3, 3, 0]$ |
Invariants of the Galois closure
Galois group: | $F_8:C_3$ (as 8T36) |
Inertia group: | $F_8$ (as 8T25) |
Wild inertia group: | $C_2^3$ |
Unramified degree: | $3$ |
Tame degree: | $7$ |
Wild slopes: | $[10/7, 10/7, 10/7]$ |
Galois mean slope: | $19/14$ |
Galois splitting model: | $x^{8} - 2 x^{7} + 8 x^{6} - 2 x^{5} + 14 x^{4} + 10 x^{3} + 28 x^{2} + 10 x + 1$ |