Defining polynomial
\(x^{8} + 10 x^{4} - 25\) |
Invariants
Base field: | $\Q_{5}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $4$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{5}$ |
Root number: | $-1$ |
$\card{ \Gal(K/\Q_{ 5 }) }$: | $8$ |
This field is Galois and abelian over $\Q_{5}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{5}(\sqrt{2})$, $\Q_{5}(\sqrt{5})$, $\Q_{5}(\sqrt{5\cdot 2})$, 5.4.2.1, 5.4.3.4, 5.4.3.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{5}(\sqrt{2})$ $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{2} + 4 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{4} + 5 t + 15 \) $\ \in\Q_{5}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{3} + 4z^{2} + z + 4$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_2\times C_4$ (as 8T2) |
Inertia group: | Intransitive group isomorphic to $C_4$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $4$ |
Wild slopes: | None |
Galois mean slope: | $3/4$ |
Galois splitting model: | $x^{8} + 20 x^{6} + 110 x^{4} + 200 x^{2} + 100$ |