Properties

Label 5.11.10.1
Base \(\Q_{5}\)
Degree \(11\)
e \(11\)
f \(1\)
c \(10\)
Galois group $C_{11}:C_5$ (as 11T3)

Related objects

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Defining polynomial

\( x^{11} - 5 \)

Invariants

Base field: $\Q_{5}$
Degree $d$ : $11$
Ramification exponent $e$ : $11$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $10$
Discriminant root field: $\Q_{5}$
Root number: $1$
$|\Aut(K/\Q_{ 5 })|$: $1$
This field is not Galois over $\Q_{5}$.

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 5 }$.

Unramified/totally ramified tower

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

Invariants of the Galois closure

Galois group:$C_{11}:C_5$ (as 11T3)
Inertia group:$C_{11}$
Unramified degree:$5$
Tame degree:$11$
Wild slopes:None
Galois mean slope:$10/11$
Galois splitting model:$x^{11} - 55 x^{9} + 1100 x^{7} - 9625 x^{5} + 34375 x^{3} - 34375 x - 12675$