Base \(\Q_{19}\)
Degree \(11\)
e \(11\)
f \(1\)
c \(10\)
Galois group $F_{11}$ (as 11T4)

Related objects

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Defining polynomial

\(x^{11} - 19\)  Toggle raw display


Base field: $\Q_{19}$
Degree $d$: $11$
Ramification exponent $e$: $11$
Residue field degree $f$: $1$
Discriminant exponent $c$: $10$
Discriminant root field: $\Q_{19}(\sqrt{2})$
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:\( x^{11} - 19 \)  Toggle raw display

Invariants of the Galois closure

Galois group:$F_{11}$ (as 11T4)
Inertia group:$C_{11}$
Unramified degree:$10$
Tame degree:$11$
Wild slopes:None
Galois mean slope:$10/11$
Galois splitting model:$x^{11} - 19$  Toggle raw display