Defining polynomial
| \( x^{8} + 4 x^{7} + 4 \) |
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$ : | $8$ |
| Ramification exponent $e$ : | $8$ |
| Residue field degree $f$ : | $1$ |
| Discriminant exponent $c$ : | $14$ |
| Discriminant root field: | $\Q_{2}$ |
| 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: | \( x^{8} + 1618 x^{7} - 6460 x^{6} + 10736 x^{5} - 4806 x^{4} + 92 x^{3} + 14840 x^{2} - 8760 x + 4798 \) |