Properties

Label 7.1.0.1
Base \(\Q_{7}\)
Degree \(1\)
e \(1\)
f \(1\)
c \(0\)
Galois group Trivial (as 1T1)

Related objects

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Defining polynomial

\( x + 2 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $1$
Ramification exponent $e$ : $1$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $0$
Discriminant root field: $\Q_{7}$
Root number: $1$
$|\Gal(K/\Q_{ 7 })|$: $1$
This field is Galois and abelian over $\Q_{7}$.

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 7 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{7}$
Relative Eisenstein polynomial:\( x - 7 \)

Invariants of the Galois closure

Galois group:$C_1$ (as 1T1)
Inertia group:Trivial
Unramified degree:$1$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x + 2$