Defining polynomial
\(x^{10} - 7 x + 19\)  |
Invariants
| Base field: | $\Q_{43}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $10$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{43}(\sqrt{2})$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 43 })|$: | $10$ |
| This field is Galois and abelian over $\Q_{43}.$ | |
Intermediate fields
| $\Q_{43}(\sqrt{2})$, 43.5.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 43.10.0.1 $\cong \Q_{43}(t)$ where $t$ is a root of \( x^{10} - 7 x + 19 \) |
| Relative Eisenstein polynomial: | \( x - 43 \)$\ \in\Q_{43}(t)[x]$ |