Properties

Label 7.4.2.2
Base \(\Q_{7}\)
Degree \(4\)
e \(2\)
f \(2\)
c \(2\)
Galois group $C_4$

Related objects

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Defining polynomial

\( x^{4} - 7 x^{2} + 147 \)

Invariants

Base field: $\Q_{ 7 }$
Degree $d$ : $4$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $2$
Discriminant root field: $\Q_{7}(\sqrt{*})$
Root number: $-1$
$|\Gal(K/\Q_{ 7 })|$: $4$
This field is Galois and abelian over $\Q_{7}$.

Intermediate fields

7.2.0.1

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

Unramified/totally ramified tower

Unramified subfield:\( t^{2} - t + 3 \)
Relative Eisenstein polynomial:\( y^{2} - 7 t \)

Invariants of the Galois closure

Galois group:$C_4$
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$2$
Tame degree:$2$
Wild slopes:None
Galois Mean Slope:$1/2$
Galois Splitting Model:\( x^{4} - x^{3} - 9 x^{2} + 9 x + 11 \)