Defining polynomial
|
\(x^{10} + 10 x^{5} + 5 x^{4} + 10\)
|
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $10$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $13$ |
| Discriminant root field: | $\Q_{5}(\sqrt{5\cdot 2})$ |
| Root number: | $-1$ |
| $\card{ \Aut(K/\Q_{ 5 }) }$: | $5$ |
| This field is not Galois over $\Q_{5}.$ | |
| Visible slopes: | $[3/2]$ |
Intermediate fields
| $\Q_{5}(\sqrt{5\cdot 2})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{5}$ |
| Relative Eisenstein polynomial: |
\( x^{10} + 10 x^{5} + 5 x^{4} + 10 \)
|
Ramification polygon
| Residual polynomials: | $2z^{4} + 3$,$z^{5} + 2$ |
| Associated inertia: | $1$,$1$ |
| Indices of inseparability: | $[4, 0]$ |