Properties

Label 7.4.3.2
Base \(\Q_{7}\)
Degree \(4\)
e \(4\)
f \(1\)
c \(3\)
Galois group $D_{4}$ (as 4T3)

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Defining polynomial

\(x^{4} - 7\) Copy content Toggle raw display

Invariants

Base field: $\Q_{7}$
Degree $d$: $4$
Ramification exponent $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $3$
Discriminant root field: $\Q_{7}(\sqrt{7\cdot 3})$
Root number: $-i$
$\card{ \Aut(K/\Q_{ 7 }) }$: $2$
This field is not Galois over $\Q_{7}.$
Visible slopes:None

Intermediate fields

$\Q_{7}(\sqrt{7})$

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{7}$
Relative Eisenstein polynomial: \( x^{4} - 7 \) Copy content Toggle raw display
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:$x^{4} - 7$