Properties

Label 191.5.4.2
Base \(\Q_{191}\)
Degree \(5\)
e \(5\)
f \(1\)
c \(4\)
Galois group $C_5$ (as 5T1)

Related objects

Learn more about

Defining polynomial

\( x^{5} + 382 \)

Invariants

Base field: $\Q_{191}$
Degree $d$ : $5$
Ramification exponent $e$ : $5$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $4$
Discriminant root field: $\Q_{191}$
Root number: $1$
$|\Gal(K/\Q_{ 191 })|$: $5$
This field is Galois and abelian over $\Q_{191}$.

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 191 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{191}$
Relative Eisenstein polynomial:\( x^{5} + 382 \)

Invariants of the Galois closure

Galois group:$C_5$ (as 5T1)
Inertia group:$C_5$
Unramified degree:$1$
Tame degree:$5$
Wild slopes:None
Galois mean slope:$4/5$
Galois splitting model:\( x^{5} - x^{4} - 840 x^{3} - 1933 x^{2} + 96209 x - 110243 \)