Defining polynomial
\(x^{4} + 17\) |
Invariants
Base field: | $\Q_{17}$ |
Degree $d$: | $4$ |
Ramification exponent $e$: | $4$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $3$ |
Discriminant root field: | $\Q_{17}(\sqrt{17})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 17 }) }$: | $4$ |
This field is Galois and abelian over $\Q_{17}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{17}(\sqrt{17})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{17}$ |
Relative Eisenstein polynomial: | \( x^{4} + 17 \) |
Ramification polygon
Residual polynomials: | $z^{3} + 4z^{2} + 6z + 4$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |