Defining polynomial
\(x^{5} + x + 4\) |
Invariants
Base field: | $\Q_{7}$ |
Degree $d$: | $5$ |
Ramification exponent $e$: | $1$ |
Residue field degree $f$: | $5$ |
Discriminant exponent $c$: | $0$ |
Discriminant root field: | $\Q_{7}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 7 }) }$: | $5$ |
This field is Galois and abelian over $\Q_{7}.$ | |
Visible slopes: | None |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q_{ 7 }$. |
Unramified/totally ramified tower
Unramified subfield: | 7.5.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{5} + x + 4 \) |
Relative Eisenstein polynomial: | \( x - 7 \) $\ \in\Q_{7}(t)[x]$ |
Ramification polygon
The ramification polygon is trivial for unramified extensions.