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

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Defining polynomial

\(x^{4} + 19\) Copy content Toggle raw display


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: $-i$
$\card{ \Aut(K/\Q_{ 19 }) }$: $2$
This field is not Galois over $\Q_{19}.$
Visible slopes:None

Intermediate fields

$\Q_{19}(\sqrt{19\cdot 2})$

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: \( x^{4} + 19 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{3} + 4z^{2} + 6z + 4$
Associated inertia:$2$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$D_4$ (as 4T3)
Inertia group:$C_4$ (as 4T1)
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$4$
Wild slopes:None
Galois mean slope:$3/4$
Galois splitting model:$x^{4} + 76$