Defining polynomial
| \( x^{7} - 4 x + 5 \) |
Invariants
| Base field: | $\Q_{37}$ |
| Degree $d$ : | $7$ |
| Ramification exponent $e$ : | $1$ |
| Residue field degree $f$ : | $7$ |
| Discriminant exponent $c$ : | $0$ |
| Discriminant root field: | $\Q_{37}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 37 })|$: | $7$ |
| This field is Galois and abelian over $\Q_{37}$. | |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 37 }$. |
Unramified/totally ramified tower
| Unramified subfield: | 37.7.0.1 $\cong \Q_{37}(t)$ where $t$ is a root of \( x^{7} - 4 x + 5 \) |
| Relative Eisenstein polynomial: | $ x - 37 \in\Q_{37}(t)[x]$ |