Properties

Label 2.2.2.2
Base \(\Q_{2}\)
Degree \(2\)
e \(2\)
f \(1\)
c \(2\)
Galois group $C_2$

Related objects

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Defining polynomial

\( x^{2} + 2 x - 2 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $2$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $2$
Discriminant root field: $\Q_{2}(\sqrt{-*})$
Root number: $i$
$|\Gal(K/\Q_{ 2 })|$: $2$
This field is Galois and abelian 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:\( x^{2} + 2 x - 2 \)

Invariants of the Galois closure

Galois group:$C_2$
Inertia group:$C_2$
Unramified degree:$1$
Tame degree:$1$
Wild slopes:[2]
Galois mean slope:$1$
Global splitting model:\( x^{2} + 2 x - 2 \)