Properties

Label 139.4.0.1
Base \(\Q_{139}\)
Degree \(4\)
e \(1\)
f \(4\)
c \(0\)
Galois group $C_4$ (as 4T1)

Related objects

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Defining polynomial

\(x^{4} - x + 2\)  Toggle raw display

Invariants

Base field: $\Q_{139}$
Degree $d$: $4$
Ramification exponent $e$: $1$
Residue field degree $f$: $4$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{139}(\sqrt{2})$
Root number: $1$
$|\Gal(K/\Q_{ 139 })|$: $4$
This field is Galois and abelian over $\Q_{139}.$

Intermediate fields

$\Q_{139}(\sqrt{2})$

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:139.4.0.1 $\cong \Q_{139}(t)$ where $t$ is a root of \( x^{4} - x + 2 \)  Toggle raw display
Relative Eisenstein polynomial:\( x - 139 \)$\ \in\Q_{139}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_4$ (as 4T1)
Inertia group:trivial
Unramified degree:$4$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{4} - x^{3} - 6 x^{2} + x + 1$