Properties

Label 2.4.6.7
Base \(\Q_{2}\)
Degree \(4\)
e \(4\)
f \(1\)
c \(6\)
Galois group $A_4$ (as 4T4)

Related objects

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Defining polynomial

\(x^{4} + 2 x^{3} + 2 x^{2} + 2\)  Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $4$
Ramification exponent $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{2}$
Root number: $-1$
$\card{ \Aut(K/\Q_{ 2 }) }$: $1$
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:\( x^{4} + 2 x^{3} + 2 x^{2} + 2 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$A_4$ (as 4T4)
Inertia group:$C_2^2$
Wild inertia group:$C_2^2$
Unramified degree:$3$
Tame degree:$1$
Wild slopes:[2, 2]
Galois mean slope:$3/2$
Galois splitting model:$x^{4} + 2 x^{3} + 2 x^{2} + 2$  Toggle raw display

Additional information

This is the only degree $4$ extension of $\Q_p$, for any $p$, which has Galois group $A_4$.