Properties

Label 19.8.0.1
Base \(\Q_{19}\)
Degree \(8\)
e \(1\)
f \(8\)
c \(0\)
Galois group $C_8$ (as 8T1)

Related objects

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Defining polynomial

\( x^{8} - x + 2 \)

Invariants

Base field: $\Q_{19}$
Degree $d$ : $8$
Ramification exponent $e$ : $1$
Residue field degree $f$ : $8$
Discriminant exponent $c$ : $0$
Discriminant root field: $\Q_{19}(\sqrt{*})$
Root number: $1$
$|\Gal(K/\Q_{ 19 })|$: $8$
This field is Galois and abelian over $\Q_{19}$.

Intermediate fields

$\Q_{19}(\sqrt{*})$, 19.4.0.1

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

Unramified/totally ramified tower

Unramified subfield:19.8.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{8} - x + 2 \)
Relative Eisenstein polynomial:$ x - 19 \in\Q_{19}(t)[x]$

Invariants of the Galois closure

Galois group:$C_8$ (as 8T1)
Inertia group:Trivial
Unramified degree:$8$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{8} - x^{7} + 3 x^{6} - 11 x^{5} + 44 x^{4} + 53 x^{3} + 153 x^{2} + 160 x + 59$