Properties

Label 3.2.1.1
Base \(\Q_{3}\)
Degree \(2\)
e \(2\)
f \(1\)
c \(1\)
Galois group $C_2$ (as 2T1)

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Defining polynomial

\(x^{2} - 3\) Copy content Toggle raw display

Invariants

Base field: $\Q_{3}$
Degree $d$: $2$
Ramification exponent $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $1$
Discriminant root field: $\Q_{3}(\sqrt{3})$
Root number: $-i$
$\card{ \Gal(K/\Q_{ 3 }) }$: $2$
This field is Galois and abelian over $\Q_{3}.$
Visible slopes:None

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 3 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{3}$
Relative Eisenstein polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_2$ (as 2T1)
Inertia group:$C_2$ (as 2T1)
Wild inertia group:$C_1$
Unramified degree:$1$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{2} - 3$