Defining polynomial
|
\(x^{10} + 113 x^{5} + 47 x^{4} + 173 x^{3} + 74 x^{2} + 156 x + 19\)
|
Invariants
| Base field: | $\Q_{191}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $10$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{191}(\sqrt{7})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 191 }) }$: | $10$ |
| This field is Galois and abelian over $\Q_{191}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{191}(\sqrt{7})$, 191.5.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 191.10.0.1 $\cong \Q_{191}(t)$ where $t$ is a root of
\( x^{10} + 113 x^{5} + 47 x^{4} + 173 x^{3} + 74 x^{2} + 156 x + 19 \)
|
| Relative Eisenstein polynomial: |
\( x - 191 \)
$\ \in\Q_{191}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.