Properties

Label 7.4.3.1
Base \(\Q_{7}\)
Degree \(4\)
e \(4\)
f \(1\)
c \(3\)
Galois group $D_4$

Related objects

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Defining polynomial

\( x^{4} + 14 \)

Invariants

Base field: $\Q_{ 7 }$
Degree $d$ : $4$
Ramification exponent $e$ : $4$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $3$
Discriminant root field: $\Q_{7}(\sqrt{7})$
Root number: $i$
$|\Aut(K/\Q_{ 7 })|$: $2$
This field is not Galois over $\Q_{7}$.

Intermediate fields

7.2.1.2

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

Unramified/totally ramified tower

Unramified subfield:\( t + 2 \)
Relative Eisenstein polynomial:\( y^{4} - 7 t \)

Invariants of the Galois closure

Galois group:$D_4$
Inertia group:$C_4$
Unramified degree:$2$
Tame degree:$4$
Wild slopes:None
Galois Mean Slope:$3/4$
Galois Splitting Model:\( x^{4} + 14 \)