Defining polynomial
| \( x^{5} - 89 \) |
Invariants
| Base field: | $\Q_{89}$ |
| Degree $d$ : | $5$ |
| Ramification exponent $e$ : | $5$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $4$ |
| Discriminant root field: | $\Q_{89}$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 89 })|$: | $1$ |
| This field is not Galois over $\Q_{89}$. | |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 89 }$. |
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{89}$ |
| Relative Eisenstein polynomial: | \( x^{5} - 89 \) |