Properties

Label 7.4.2.1
Base \(\Q_{7}\)
Degree \(4\)
e \(2\)
f \(2\)
c \(2\)
Galois group $V_4$ (as 4T2)

Related objects

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Defining polynomial

\( x^{4} + 35 x^{2} + 441 \)

Invariants

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

Intermediate fields

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

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{7}(\sqrt{*})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} - x + 3 \)
Relative Eisenstein polynomial:$ x^{2} - 7 t^{2} \in\Q_{7}(t)[x]$

Invariants of the Galois closure

Galois group:$C_2^2$ (as 4T2)
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:\( x^{4} + 35 x^{2} + 441 \)