Base \(\Q_{5}\)
Degree \(3\)
e \(1\)
f \(3\)
c \(0\)
Galois group $C_3$ (as 3T1)

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Defining polynomial

\(x^{3} - x + 2\) Copy content Toggle raw display


Base field: $\Q_{5}$
Degree $d$: $3$
Ramification exponent $e$: $1$
Residue field degree $f$: $3$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{5}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 5 }) }$: $3$
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: $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{3} - x + 2 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x - 5 \) $\ \in\Q_{5}(t)[x]$ Copy content Toggle raw display
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_3$ (as 3T1)
Inertia group:trivial
Wild inertia group:$C_1$
Unramified degree:$3$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{3} - x^{2} - 2 x + 1$