Defining polynomial
| \( x^{4} - 139 \) |
Invariants
| Base field: | $\Q_{139}$ |
| Degree $d$ : | $4$ |
| Ramification exponent $e$ : | $4$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $3$ |
| Discriminant root field: | $\Q_{139}(\sqrt{139*})$ |
| Root number: | $i$ |
| $|\Aut(K/\Q_{ 139 })|$: | $2$ |
| This field is not Galois over $\Q_{139}$. | |
Intermediate fields
| $\Q_{139}(\sqrt{139})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{139}$ |
| Relative Eisenstein polynomial: | \( x^{4} - 139 \) |
Invariants of the Galois closure
| Galois group: | $D_4$ (as 4T3) |
| Inertia group: | $C_4$ |
| Unramified degree: | $2$ |
| Tame degree: | $4$ |
| Wild slopes: | None |
| Galois mean slope: | $3/4$ |
| Galois splitting model: | $x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 117$ |