Properties

Label 31.2.1.2
Base \(\Q_{31}\)
Degree \(2\)
e \(2\)
f \(1\)
c \(1\)
Galois group $C_2$ (as 2T1)

Related objects

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Defining polynomial

\( x^{2} + 217 \)

Invariants

Base field: $\Q_{31}$
Degree $d$: $2$
Ramification exponent $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $1$
Discriminant root field: $\Q_{31}(\sqrt{31*})$
Root number: $i$
$|\Gal(K/\Q_{ 31 })|$: $2$
This field is Galois and abelian over $\Q_{31}.$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 31 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{31}$
Relative Eisenstein polynomial:\( x^{2} + 217 \)

Invariants of the Galois closure

Galois group:$C_2$ (as 2T1)
Inertia group:$C_2$
Unramified degree:$1$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{2} + 217$