Defining polynomial
\(x^{8} + x^{4} + x^{3} + x^{2} + 1\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $1$ |
Residue field degree $f$: | $8$ |
Discriminant exponent $c$: | $0$ |
Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 2 }) }$: | $8$ |
This field is Galois and abelian over $\Q_{2}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{2}(\sqrt{5})$, 2.4.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 2.8.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{8} + x^{4} + x^{3} + x^{2} + 1 \) |
Relative Eisenstein polynomial: | \( x - 2 \) $\ \in\Q_{2}(t)[x]$ |
Ramification polygon
The ramification polygon is trivial for unramified extensions.
Invariants of the Galois closure
Galois group: | $C_8$ (as 8T1) |
Inertia group: | trivial |
Wild inertia group: | $C_1$ |
Unramified degree: | $8$ |
Tame degree: | $1$ |
Wild slopes: | None |
Galois mean slope: | $0$ |
Galois splitting model: | Does not exist |
Additional information
This octic field is not the $2$-completion of an octic extension of $\Q$ with Galois group $C_8$. It has minimal degree for this phenomenon, being one of several such fields with Galois group $C_8$.