Defining polynomial
\( x^{8} + 35 x^{4} + 441 \) |
Invariants
Base field: | $\Q_{7}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $4$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{7}$ |
Root number: | $1$ |
$|\Gal(K/\Q_{ 7 })|$: | $8$ |
This field is Galois over $\Q_{7}.$ |
Intermediate fields
$\Q_{7}(\sqrt{3})$, $\Q_{7}(\sqrt{7})$, $\Q_{7}(\sqrt{7\cdot 3})$, 7.4.2.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{7}(\sqrt{3})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} - x + 3 \) |
Relative Eisenstein polynomial: | $ x^{4} - 7 t^{2} \in\Q_{7}(t)[x]$ |
Invariants of the Galois closure
Galois group: | $Q_8$ (as 8T5) |
Inertia group: | Intransitive group isomorphic to $C_4$ |
Unramified degree: | $2$ |
Tame degree: | $4$ |
Wild slopes: | None |
Galois mean slope: | $3/4$ |
Galois splitting model: | $x^{8} - 3 x^{7} + 22 x^{6} - 60 x^{5} + 201 x^{4} - 450 x^{3} + 1528 x^{2} - 3069 x + 4561$ |