Defining polynomial
\(x^{7} + 21 x^{4} + 7\) |
Invariants
Base field: | $\Q_{7}$ |
Degree $d$: | $7$ |
Ramification exponent $e$: | $7$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $10$ |
Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 7 }) }$: | $1$ |
This field is not Galois over $\Q_{7}.$ | |
Visible slopes: | $[5/3]$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 7 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{7}$ |
Relative Eisenstein polynomial: | \( x^{7} + 21 x^{4} + 7 \) |
Ramification polygon
Residual polynomials: | $z^{2} + 2$ |
Associated inertia: | $2$ |
Indices of inseparability: | $[4, 0]$ |
Invariants of the Galois closure
Galois group: | $F_7$ (as 7T4) |
Inertia group: | $C_7:C_3$ (as 7T3) |
Wild inertia group: | $C_7$ |
Unramified degree: | $2$ |
Tame degree: | $3$ |
Wild slopes: | $[5/3]$ |
Galois mean slope: | $32/21$ |
Galois splitting model: | $x^{7} - 70 x^{5} - 77 x^{4} + 1169 x^{3} + 2058 x^{2} - 1148 x - 1005$ |