Properties

Label 3.1.0.1
Base \(\Q_{3}\)
Degree \(1\)
e \(1\)
f \(1\)
c \(0\)
Galois group Trivial (as 1T1)

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Defining polynomial

\(x + 1\) Copy content Toggle raw display

Invariants

Base field: $\Q_{3}$
Degree $d$: $1$
Ramification exponent $e$: $1$
Residue field degree $f$: $1$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{3}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 3 }) }$: $1$
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: \( x - 3 \) Copy content Toggle raw display
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_1$ (as 1T1)
Inertia group:$C_1$ (as 1T1)
Wild inertia group:$C_1$
Unramified degree:$1$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model: $x + 1$ Copy content Toggle raw display