Defining polynomial
\(x^{7} - x + 8\) ![]() |
Invariants
Base field: | $\Q_{23}$ |
Degree $d$: | $7$ |
Ramification exponent $e$: | $1$ |
Residue field degree $f$: | $7$ |
Discriminant exponent $c$: | $0$ |
Discriminant root field: | $\Q_{23}$ |
Root number: | $1$ |
$|\Gal(K/\Q_{ 23 })|$: | $7$ |
This field is Galois and abelian over $\Q_{23}.$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 23 }$. |
Unramified/totally ramified tower
Unramified subfield: | 23.7.0.1 $\cong \Q_{23}(t)$ where $t$ is a root of \( x^{7} - x + 8 \) ![]() |
Relative Eisenstein polynomial: | \( x - 23 \)$\ \in\Q_{23}(t)[x]$ ![]() |