Defining polynomial
\(x^{10} + 15 x^{4} + 5\) |
Invariants
Base field: | $\Q_{5}$ |
Degree $d$: | $10$ |
Ramification exponent $e$: | $10$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $13$ |
Discriminant root field: | $\Q_{5}(\sqrt{5})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 5 }) }$: | $10$ |
This field is Galois over $\Q_{5}.$ | |
Visible slopes: | $[3/2]$ |
Intermediate fields
$\Q_{5}(\sqrt{5})$, 5.5.6.2 x5 |
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: | \( x^{10} + 15 x^{4} + 5 \) |
Ramification polygon
Residual polynomials: | $2z^{4} + 3$,$z^{5} + 2$ |
Associated inertia: | $1$,$1$ |
Indices of inseparability: | $[4, 0]$ |