Properties

Label 19.3.0.1
Base \(\Q_{19}\)
Degree \(3\)
e \(1\)
f \(3\)
c \(0\)
Galois group $C_3$ (as 3T1)

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Defining polynomial

\(x^{3} + 4 x + 17\) Copy content Toggle raw display

Invariants

Base field: $\Q_{19}$
Degree $d$: $3$
Ramification exponent $e$: $1$
Residue field degree $f$: $3$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{19}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 19 }) }$: $3$
This field is Galois and abelian over $\Q_{19}.$
Visible slopes:None

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 19 }$.

Unramified/totally ramified tower

Unramified subfield:19.3.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{3} + 4 x + 17 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x - 19 \) $\ \in\Q_{19}(t)[x]$ Copy content Toggle raw display

Ramification polygon

The ramification polygon is trivial for unramified extensions.

Invariants of the Galois closure

Galois group: $C_3$ (as 3T1)
Inertia group: trivial
Wild inertia group: $C_1$
Unramified degree: $3$
Tame degree: $1$
Wild slopes: None
Galois mean slope: $0$
Galois splitting model:$x^{3} - x^{2} - 2 x + 1$