Defining polynomial
\(x^{8} + 245 x^{4} - 1372 x^{2} + 7203\) |
Invariants
Base field: | $\Q_{7}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $4$ |
Discriminant exponent $c$: | $4$ |
Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 7 }) }$: | $8$ |
This field is Galois and abelian over $\Q_{7}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{7}(\sqrt{3})$, 7.4.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 7.4.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{4} + 5 x^{2} + 4 x + 3 \) |
Relative Eisenstein polynomial: | \( x^{2} + 7 t \) $\ \in\Q_{7}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 2$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_8$ (as 8T1) |
Inertia group: | Intransitive group isomorphic to $C_2$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $4$ |
Tame degree: | $2$ |
Wild slopes: | None |
Galois mean slope: | $1/2$ |
Galois splitting model: | $x^{8} - x^{7} + 27 x^{6} - 28 x^{5} + 151 x^{4} - 350 x^{3} + 500 x^{2} - 846 x + 1157$ |