Properties

Label 5.7.0.1
Base \(\Q_{5}\)
Degree \(7\)
e \(1\)
f \(7\)
c \(0\)
Galois group $C_7$ (as 7T1)

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Defining polynomial

\(x^{7} + 3 x + 3\) Copy content Toggle raw display

Invariants

Base field: $\Q_{5}$
Degree $d$: $7$
Ramification exponent $e$: $1$
Residue field degree $f$: $7$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{5}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 5 }) }$: $7$
This field is Galois and abelian over $\Q_{5}.$
Visible slopes:None

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 5 }$.

Unramified/totally ramified tower

Unramified subfield:5.7.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{7} + 3 x + 3 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x - 5 \) $\ \in\Q_{5}(t)[x]$ Copy content Toggle raw display

Ramification polygon

The ramification polygon is trivial for unramified extensions.

Invariants of the Galois closure

Galois group:$C_7$ (as 7T1)
Inertia group:trivial
Wild inertia group:$C_1$
Unramified degree:$7$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$