Defining polynomial
\(x^{4} - 1806 x^{2} + 5547\) |
Invariants
Base field: | $\Q_{43}$ |
Degree $d$: | $4$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $2$ |
Discriminant root field: | $\Q_{43}(\sqrt{2})$ |
Root number: | $-1$ |
$\card{ \Gal(K/\Q_{ 43 }) }$: | $4$ |
This field is Galois and abelian over $\Q_{43}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{43}(\sqrt{2})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{43}(\sqrt{2})$ $\cong \Q_{43}(t)$ where $t$ is a root of \( x^{2} + 42 x + 3 \) |
Relative Eisenstein polynomial: | \( x^{2} + 43 t \) $\ \in\Q_{43}(t)[x]$ |
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: | $x^{4} - x^{3} - 54 x^{2} + 54 x + 551$ |