Properties

Label 3.11.0.1
Base \(\Q_{3}\)
Degree \(11\)
e \(1\)
f \(11\)
c \(0\)
Galois group $C_{11}$

Related objects

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Defining polynomial

\( x^{11} + x^{2} - x + 1 \)

Invariants

Base field: $\Q_{ 3 }$
Degree $d$ : $11$
Ramification exponent $e$ : $1$
Residue field degree $f$ : $11$
Discriminant exponent $c$ : $0$
Discriminant root field: $\Q_{3}$
Root number: $1$
$|\Gal(K/\Q_{ 3 })|$: $11$
This field is Galois and abelian over $\Q_{3}$.

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 3 }$.

Unramified/totally ramified tower

Unramified subfield:\( t^{11} + t^{2} - t + 1 \)
Relative Eisenstein polynomial:\( y - 3 \)

Invariants of the Galois closure

Galois group:$C_{11}$
Inertia group:Trivial
Unramified degree:$11$
Tame degree:$1$
Wild slopes:None
Galois Mean Slope:$0$
Galois Splitting Model:\( x^{11} - x^{10} - 10 x^{9} + 9 x^{8} + 36 x^{7} - 28 x^{6} - 56 x^{5} + 35 x^{4} + 35 x^{3} - 15 x^{2} - 6 x + 1 \)