Properties

Label 19.5.4.1
Base \(\Q_{19}\)
Degree \(5\)
e \(5\)
f \(1\)
c \(4\)
Galois group $D_{5}$ (as 5T2)

Related objects

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Defining polynomial

\( x^{5} - 19 \)

Invariants

Base field: $\Q_{19}$
Degree $d$ : $5$
Ramification exponent $e$ : $5$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $4$
Discriminant root field: $\Q_{19}$
Root number: $1$
$|\Aut(K/\Q_{ 19 })|$: $1$
This field is not Galois over $\Q_{19}$.

Intermediate fields

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{19}$
Relative Eisenstein polynomial:\( x^{5} - 19 \)

Invariants of the Galois closure

Galois group:$D_5$ (as 5T2)
Inertia group:$C_5$
Unramified degree:$2$
Tame degree:$5$
Wild slopes:None
Galois mean slope:$4/5$
Galois splitting model:$x^{5} - 2 x^{4} - 6 x^{3} + 10 x^{2} + 17 x - 12$