Defining polynomial
\(x^{6} + 37\) |
Invariants
Base field: | $\Q_{37}$ |
Degree $d$: | $6$ |
Ramification exponent $e$: | $6$ |
Residue field degree $f$: | $1$ |
Discriminant exponent $c$: | $5$ |
Discriminant root field: | $\Q_{37}(\sqrt{37})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 37 }) }$: | $6$ |
This field is Galois and abelian over $\Q_{37}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{37}(\sqrt{37})$, 37.3.2.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{37}$ |
Relative Eisenstein polynomial: | \( x^{6} + 37 \) |
Ramification polygon
Residual polynomials: | $z^{5} + 6z^{4} + 15z^{3} + 20z^{2} + 15z + 6$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |