Base \(\Q_{5}\)
Degree \(6\)
e \(1\)
f \(6\)
c \(0\)
Galois group $C_6$ (as 6T1)

Related objects

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Defining polynomial

\(x^{6} - x + 2\)  Toggle raw display


Base field: $\Q_{5}$
Degree $d$: $6$
Ramification exponent $e$: $1$
Residue field degree $f$: $6$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{5}(\sqrt{2})$
Root number: $1$
$|\Gal(K/\Q_{ 5 })|$: $6$
This field is Galois and abelian over $\Q_{5}.$

Intermediate fields


Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield: $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{6} - x + 2 \)  Toggle raw display
Relative Eisenstein polynomial:\( x - 5 \)$\ \in\Q_{5}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_6$ (as 6T1)
Inertia group:trivial
Wild inertia group:$C_1$
Unramified degree:$6$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$