Properties

Label 11.5.4.3
Base \(\Q_{11}\)
Degree \(5\)
e \(5\)
f \(1\)
c \(4\)
Galois group $C_5$ (as 5T1)

Related objects

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Defining polynomial

\( x^{5} + 33 \)

Invariants

Base field: $\Q_{11}$
Degree $d$: $5$
Ramification exponent $e$: $5$
Residue field degree $f$: $1$
Discriminant exponent $c$: $4$
Discriminant root field: $\Q_{11}$
Root number: $1$
$|\Gal(K/\Q_{ 11 })|$: $5$
This field is Galois and abelian over $\Q_{11}.$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 11 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{11}$
Relative Eisenstein polynomial:\( x^{5} + 33 \)

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:$x^{5} - 110 x^{3} - 605 x^{2} - 990 x - 451$