Defining polynomial
| \( x^{6} - x + 21 \) |
Invariants
| Base field: | $\Q_{139}$ |
| Degree $d$ : | $6$ |
| Ramification exponent $e$ : | $1$ |
| Residue field degree $f$ : | $6$ |
| Discriminant exponent $c$ : | $0$ |
| Discriminant root field: | $\Q_{139}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 139 })|$: | $6$ |
| This field is Galois and abelian over $\Q_{139}$. | |
Intermediate fields
| $\Q_{139}(\sqrt{*})$, 139.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 139.6.0.1 $\cong \Q_{139}(t)$ where $t$ is a root of \( x^{6} - x + 21 \) |
| Relative Eisenstein polynomial: | $ x - 139 \in\Q_{139}(t)[x]$ |