Properties

Label 89.1.0.1
Base \(\Q_{89}\)
Degree \(1\)
e \(1\)
f \(1\)
c \(0\)
Galois group Trivial (as 1T1)

Related objects

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Defining polynomial

\(x + 3\)  Toggle raw display

Invariants

Base field: $\Q_{89}$
Degree $d$: $1$
Ramification exponent $e$: $1$
Residue field degree $f$: $1$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{89}$
Root number: $1$
$|\Gal(K/\Q_{ 89 })|$: $1$
This field is Galois and abelian over $\Q_{89}.$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 89 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{89}$
Relative Eisenstein polynomial:\( x - 89 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$C_1$ (as 1T1)
Inertia group:Trivial
Wild inertia group:$C_1$
Unramified degree:$1$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x + 3$  Toggle raw display