Defining polynomial
| \( x^{4} - 801 \) |
Invariants
| Base field: | $\Q_{89}$ |
| Degree $d$ : | $4$ |
| Ramification exponent $e$ : | $4$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $3$ |
| Discriminant root field: | $\Q_{89}(\sqrt{89})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 89 })|$: | $4$ |
| This field is Galois and abelian over $\Q_{89}$. | |
Intermediate fields
| $\Q_{89}(\sqrt{89})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{89}$ |
| Relative Eisenstein polynomial: | \( x^{4} - 801 \) |
Invariants of the Galois closure
| Galois group: | $C_4$ (as 4T1) |
| Inertia group: | $C_4$ |
| Unramified degree: | $1$ |
| Tame degree: | $4$ |
| Wild slopes: | None |
| Galois mean slope: | $3/4$ |
| Galois splitting model: | $x^{4} - x^{3} + 56 x^{2} + 139 x + 1165$ |