Base \(\Q_{19}\)
Degree \(4\)
e \(4\)
f \(1\)
c \(3\)
Galois group $D_{4}$ (as 4T3)

Related objects

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Defining polynomial

\( x^{4} + 76 \)


Base field: $\Q_{19}$
Degree $d$ : $4$
Ramification exponent $e$ : $4$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $3$
Discriminant root field: $\Q_{19}(\sqrt{19})$
Root number: $-i$
$|\Aut(K/\Q_{ 19 })|$: $2$
This field is not Galois over $\Q_{19}$.

Intermediate fields


Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:$\Q_{19}$
Relative Eisenstein polynomial:\( x^{4} + 76 \)

Invariants of the Galois closure

Galois group:$D_4$ (as 4T3)
Inertia group:$C_4$
Unramified degree:$2$
Tame degree:$4$
Wild slopes:None
Galois mean slope:$3/4$
Galois splitting model:$x^{4} + 76$