Properties

Label 97.2.0.1
Base \(\Q_{97}\)
Degree \(2\)
e \(1\)
f \(2\)
c \(0\)
Galois group $C_2$ (as 2T1)

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Defining polynomial

\(x^{2} + 96 x + 5\) Copy content Toggle raw display

Invariants

Base field: $\Q_{97}$
Degree $d$: $2$
Ramification exponent $e$: $1$
Residue field degree $f$: $2$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{97}(\sqrt{5})$
Root number: $1$
$\card{ \Gal(K/\Q_{ 97 }) }$: $2$
This field is Galois and abelian over $\Q_{97}.$
Visible slopes:None

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 97 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{97}(\sqrt{5})$ $\cong \Q_{97}(t)$ where $t$ is a root of \( x^{2} + 96 x + 5 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x - 97 \) $\ \in\Q_{97}(t)[x]$ Copy content Toggle raw display

Ramification polygon

The ramification polygon is trivial for unramified extensions.

Invariants of the Galois closure

Galois group:$C_2$ (as 2T1)
Inertia group:trivial
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{2} - x + 5$