Properties

Label 2.8.0.1
Base \(\Q_{2}\)
Degree \(8\)
e \(1\)
f \(8\)
c \(0\)
Galois group $C_8$ (as 8T1)

Related objects

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Defining polynomial

\(x^{8} + x^{4} + x^{3} + x + 1\)  Toggle raw display

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: $1$
$|\Gal(K/\Q_{ 2 })|$: $8$
This field is Galois and abelian over $\Q_{2}.$

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 \( x^{8} + x^{4} - x^{3} + x^{2} + 1 \)  Toggle raw display
Relative Eisenstein polynomial:\( x - 2 \)$\ \in\Q_{2}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_8$ (as 8T1)
Inertia group:trivial
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 $\Q$. It has minimal degree for this phenomenon being one of several such fields with Galois group $C_8$.