Defining polynomial
| \( x^{13} - 2 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $13$ |
| Ramification exponent $e$ : | $13$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $12$ |
| Discriminant root field: | $\Q_{2}(\sqrt{*})$ |
| Root number: | $1$ |
| $|\Aut(K/\Q_{ 2 })|$: | $1$ |
| This field is not Galois over $\Q_{2}$. | |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q_{ 2 }$. |
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{2}$ |
| Relative Eisenstein polynomial: | \( x^{13} - 2 \) |
Invariants of the Galois closure
| Galois group: | $F_{13}$ (as 13T6) |
| Inertia group: | $C_{13}$ |
| Unramified degree: | $12$ |
| Tame degree: | $13$ |
| Wild slopes: | None |
| Galois mean slope: | $12/13$ |
| Galois splitting model: | $x^{13} - 13 x^{12} + 78 x^{11} - 286 x^{10} + 715 x^{9} - 1287 x^{8} + 1716 x^{7} - 1716 x^{6} + 1287 x^{5} - 715 x^{4} + 286 x^{3} - 78 x^{2} + 13 x - 3$ |