Properties

Label 19.9.0.1
Base \(\Q_{19}\)
Degree \(9\)
e \(1\)
f \(9\)
c \(0\)
Galois group $C_9$ (as 9T1)

Related objects

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Defining polynomial

\(x^{9} + x^{2} - x + 4\)  Toggle raw display

Invariants

Base field: $\Q_{19}$
Degree $d$: $9$
Ramification exponent $e$: $1$
Residue field degree $f$: $9$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{19}$
Root number: $1$
$|\Gal(K/\Q_{ 19 })|$: $9$
This field is Galois and abelian over $\Q_{19}.$

Intermediate fields

19.3.0.1

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

Unramified/totally ramified tower

Unramified subfield:19.9.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{9} + x^{2} - x + 4 \)  Toggle raw display
Relative Eisenstein polynomial:\( x - 19 \)$\ \in\Q_{19}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_9$ (as 9T1)
Inertia group:trivial
Unramified degree:$9$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{9} - x^{8} - 16 x^{7} + 11 x^{6} + 66 x^{5} - 32 x^{4} - 73 x^{3} + 7 x^{2} + 7 x - 1$