Defining polynomial
| \( x^{15} + 375 x^{14} + 415 x^{13} + 575 x^{12} + 520 x^{11} + 378 x^{10} + 145 x^{9} + 275 x^{8} + 85 x^{7} + 545 x^{6} + 127 x^{5} + 380 x^{4} + 470 x^{3} + 615 x + 368 \) |
Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$ : | $15$ |
| Ramification exponent $e$ : | $5$ |
| Residue field degree $f$ : | $3$ |
| Discriminant exponent $c$ : | $24$ |
| Discriminant root field: | $\Q_{5}$ |
| Root number: | $1$ |
| $|\Gal(K/\Q_{ 5 })|$: | $15$ |
| This field is Galois and abelian over $\Q_{5}$. | |
Intermediate fields
| 5.3.0.1, 5.5.8.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 5.3.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{3} - x + 2 \) |
| Relative Eisenstein polynomial: | $ x^{5} + \left(35 t^{2} + 50 t + 50\right) x^{4} + \left(50 t^{2} + 75 t\right) x^{3} + \left(100 t^{2} + 100 t + 75\right) x^{2} + \left(75 t^{2} + 50 t + 100\right) x + 45 t^{2} + 35 t + 5 \in\Q_{5}(t)[x]$ |