Properties

Label 2.4.8.7
Base \(\Q_{2}\)
Degree \(4\)
e \(4\)
f \(1\)
c \(8\)
Galois group $S_4$ (as 4T5)

Related objects

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Defining polynomial

\(x^{4} + 4 x^{2} + 4 x + 2\)  Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $4$
Ramification exponent $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $8$
Discriminant root field: $\Q_{2}(\sqrt{5})$
Root number: $-1$
$|\Aut(K/\Q_{ 2 })|$: $1$
This field is not Galois over $\Q_{2}.$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 2 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{2}$
Relative Eisenstein polynomial:\( x^{4} + 4 x^{2} + 4 x + 2 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$S_4$ (as 4T5)
Inertia group:$A_4$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:[8/3, 8/3]
Galois mean slope:$13/6$
Galois splitting model:$x^{4} + 4 x^{2} + 4 x + 2$  Toggle raw display