Defining polynomial
\(x^{4} - 42 x^{2} + 147\) |
Invariants
Base field: | $\Q_{7}$ |
Degree $d$: | $4$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $2$ |
Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
Root number: | $-1$ |
$\card{ \Gal(K/\Q_{ 7 }) }$: | $4$ |
This field is Galois and abelian over $\Q_{7}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{7}(\sqrt{3})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{7}(\sqrt{3})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} + 6 x + 3 \) |
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: | $C_4$ (as 4T1) |
Inertia group: | Intransitive group isomorphic to $C_2$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $2$ |
Wild slopes: | None |
Galois mean slope: | $1/2$ |
Galois splitting model: | $x^{4} - x^{3} - 9 x^{2} + 9 x + 11$ |