Properties

Label 31.2.0.1
Base \(\Q_{31}\)
Degree \(2\)
e \(1\)
f \(2\)
c \(0\)
Galois group $C_2$ (as 2T1)

Related objects

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Defining polynomial

\(x^{2} - x + 12\)  Toggle raw display

Invariants

Base field: $\Q_{31}$
Degree $d$: $2$
Ramification exponent $e$: $1$
Residue field degree $f$: $2$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{31}(\sqrt{3})$
Root number: $1$
$|\Gal(K/\Q_{ 31 })|$: $2$
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:$\Q_{31}(\sqrt{3})$ $\cong \Q_{31}(t)$ where $t$ is a root of \( x^{2} - x + 12 \)  Toggle raw display
Relative Eisenstein polynomial:\( x - 31 \)$\ \in\Q_{31}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_2$ (as 2T1)
Inertia group:trivial
Unramified degree:$2$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{2} - x + 12$