Defining polynomial
\(x^{4} + 4 x^{2} + 72 x + 3\) |
Invariants
Base field: | $\Q_{89}$ |
Degree $d$: | $4$ |
Ramification exponent $e$: | $1$ |
Residue field degree $f$: | $4$ |
Discriminant exponent $c$: | $0$ |
Discriminant root field: | $\Q_{89}(\sqrt{3})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 89 }) }$: | $4$ |
This field is Galois and abelian over $\Q_{89}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{89}(\sqrt{3})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 89.4.0.1 $\cong \Q_{89}(t)$ where $t$ is a root of \( x^{4} + 4 x^{2} + 72 x + 3 \) |
Relative Eisenstein polynomial: | \( x - 89 \) $\ \in\Q_{89}(t)[x]$ |
Ramification polygon
The ramification polygon is trivial for unramified extensions.