Defining polynomial
\(x^{10} - 50 x^{7} + 10 x^{6} + 10 x^{5} + 175 x^{4} - 250 x^{3} - 225 x^{2} + 50 x + 25\) |
Invariants
Base field: | $\Q_{5}$ |
Degree $d$: | $10$ |
Ramification exponent $e$: | $5$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $10$ |
Discriminant root field: | $\Q_{5}(\sqrt{2})$ |
Root number: | $-1$ |
$\card{ \Aut(K/\Q_{ 5 }) }$: | $2$ |
This field is not Galois over $\Q_{5}.$ | |
Visible slopes: | $[5/4]$ |
Intermediate fields
$\Q_{5}(\sqrt{2})$, 5.5.5.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{5}(\sqrt{2})$ $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{2} + 4 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{5} + \left(15 t + 5\right) x^{2} + 5 x + 5 \) $\ \in\Q_{5}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 4$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
Galois group: | $C_2\times F_5$ (as 10T5) |
Inertia group: | Intransitive group isomorphic to $F_5$ |
Wild inertia group: | $C_5$ |
Unramified degree: | $2$ |
Tame degree: | $4$ |
Wild slopes: | $[5/4]$ |
Galois mean slope: | $23/20$ |
Galois splitting model: | $x^{10} - 3 x^{5} + 3$ |