Properties

Label 5.2.1.1
Base \(\Q_{5}\)
Degree \(2\)
e \(2\)
f \(1\)
c \(1\)
Galois group $C_2$

Related objects

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Defining polynomial

\( x^{2} - 5 \)

Invariants

Base field: $\Q_{5}$
Degree $d$ : $2$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $1$
Discriminant root field: $\Q_{5}(\sqrt{5})$
Root number: $1$
$|\Gal(K/\Q_{ 5 })|$: $2$
This field is Galois and abelian over $\Q_{5}$.

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 5 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{5}$
Relative Eisenstein polynomial:\( x^{2} - 5 \)

Invariants of the Galois closure

Galois group:$C_2$
Inertia group:$C_2$
Unramified degree:$1$
Tame degree:$2$
Wild slopes:None
Galois Mean Slope:$1/2$
Global Splitting Model:\( x^{2} - 5 \)