Properties

Label 2.2.3.3
Base \(\Q_{2}\)
Degree \(2\)
e \(2\)
f \(1\)
c \(3\)
Galois group $C_2$ (as 2T1)

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Defining polynomial

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

Invariants

Base field: $\Q_{2}$
Degree $d$: $2$
Ramification exponent $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $3$
Discriminant root field: $\Q_{2}(\sqrt{-2})$
Root number: $i$
$\card{ \Gal(K/\Q_{ 2 }) }$: $2$
This field is Galois and abelian over $\Q_{2}.$
Visible slopes:$[3]$

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: \( x^{2} + 2 \) Copy content Toggle raw display
Indices of inseparability:$[2, 0]$

Invariants of the Galois closure

Galois group:$C_2$ (as 2T1)
Inertia group:$C_2$ (as 2T1)
Wild inertia group:$C_2$
Unramified degree:$1$
Tame degree:$1$
Wild slopes:$[3]$
Galois mean slope:$3/2$
Galois splitting model:$x^{2} + 2$