Properties

Label 137.5.0.1
Base \(\Q_{137}\)
Degree \(5\)
e \(1\)
f \(5\)
c \(0\)
Galois group $C_5$ (as 5T1)

Related objects

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Defining polynomial

\( x^{5} - x + 35 \)

Invariants

Base field: $\Q_{137}$
Degree $d$: $5$
Ramification exponent $e$: $1$
Residue field degree $f$: $5$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{137}$
Root number: $1$
$|\Gal(K/\Q_{ 137 })|$: $5$
This field is Galois and abelian over $\Q_{137}.$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 137 }$.

Unramified/totally ramified tower

Unramified subfield:137.5.0.1 $\cong \Q_{137}(t)$ where $t$ is a root of \( x^{5} - x + 35 \)
Relative Eisenstein polynomial:$ x - 137 \in\Q_{137}(t)[x]$

Invariants of the Galois closure

Galois group:$C_5$ (as 5T1)
Inertia group:trivial
Unramified degree:$5$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$