Base \(\Q_{2}\)
Degree \(6\)
e \(2\)
f \(3\)
c \(6\)
Galois group $A_4$ (as 6T4)

Related objects

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Defining polynomial

\(x^{6} + x^{2} - 1\)  Toggle raw display


Base field: $\Q_{2}$
Degree $d$: $6$
Ramification exponent $e$: $2$
Residue field degree $f$: $3$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{2}$
Root number: $-1$
$|\Aut(K/\Q_{ 2 })|$: $2$
This field is not Galois over $\Q_{2}.$

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: $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{3} - x + 1 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{2} + 2 x + 2 t^{2} \)$\ \in\Q_{2}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$A_4$ (as 6T4)
Inertia group:Intransitive group isomorphic to $C_2^2$
Wild inertia group:$C_2^2$
Unramified degree:$3$
Tame degree:$1$
Wild slopes:[2, 2]
Galois mean slope:$3/2$
Galois splitting model:$x^{6} - 3 x^{2} - 1$