Defining polynomial
\(x^{5} + 20 x^{2} + 5\) |
Invariants
Base field: | $\Q_{5}$ |
Degree $d$: | $5$ |
Ramification exponent $e$: | $5$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{5}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 5 }) }$: | $1$ |
This field is not Galois over $\Q_{5}.$ | |
Visible slopes: | $[3/2]$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 5 }$. |
Unramified/totally ramified tower
Unramified subfield: | $\Q_{5}$ |
Relative Eisenstein polynomial: | \( x^{5} + 20 x^{2} + 5 \) |
Ramification polygon
Residual polynomials: | $z^{2} + 2$ |
Associated inertia: | $2$ |
Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
Galois group: | $F_5$ (as 5T3) |
Inertia group: | $D_5$ (as 5T2) |
Wild inertia group: | $C_5$ |
Unramified degree: | $2$ |
Tame degree: | $2$ |
Wild slopes: | $[3/2]$ |
Galois mean slope: | $13/10$ |
Galois splitting model: | $x^{5} - 20 x^{3} + 60 x - 32$ |