Defining polynomial
\(x^{5} + 5\) |
Invariants
Base field: | $\Q_{5}$ |
Degree $d$: | $5$ |
Ramification exponent $e$: | $5$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $9$ |
Discriminant root field: | $\Q_{5}(\sqrt{5})$ |
Root number: | $-1$ |
$\card{ \Aut(K/\Q_{ 5 }) }$: | $1$ |
This field is not Galois over $\Q_{5}.$ | |
Visible slopes: | $[9/4]$ |
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: | \( x^{5} + 5 \) |
Ramification polygon
Residual polynomials: | $z + 4$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[5, 0]$ |
Invariants of the Galois closure
Galois group: | $F_5$ (as 5T3) |
Inertia group: | $F_5$ (as 5T3) |
Wild inertia group: | $C_5$ |
Unramified degree: | $1$ |
Tame degree: | $4$ |
Wild slopes: | $[9/4]$ |
Galois mean slope: | $39/20$ |
Galois splitting model: | $x^{5} + 5$ |