Properties

Label 7.10.8.1
Base \(\Q_{7}\)
Degree \(10\)
e \(5\)
f \(2\)
c \(8\)
Galois group $F_5$ (as 10T4)

Related objects

Learn more about

Defining polynomial

\( x^{10} - 7 x^{5} + 147 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $10$
Ramification exponent $e$ : $5$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $8$
Discriminant root field: $\Q_{7}(\sqrt{*})$
Root number: $1$
$|\Aut(K/\Q_{ 7 })|$: $2$
This field is not Galois over $\Q_{7}$.

Intermediate fields

$\Q_{7}(\sqrt{*})$, 7.5.4.1

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^{5} - 7 t \in\Q_{7}(t)[x]$

Invariants of the Galois closure

Galois group:$F_5$ (as 10T4)
Inertia group:Intransitive group isomorphic to $C_5$
Unramified degree:$4$
Tame degree:$5$
Wild slopes:None
Galois mean slope:$4/5$
Galois splitting model:$x^{10} - x^{8} + 6 x^{6} - 11 x^{4} + 6 x^{2} - 5$