Properties

Label 103.3.0.1
Base \(\Q_{103}\)
Degree \(3\)
e \(1\)
f \(3\)
c \(0\)
Galois group $C_3$ (as 3T1)

Related objects

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Defining polynomial

\(x^{3} - x + 29\)  Toggle raw display

Invariants

Base field: $\Q_{103}$
Degree $d$: $3$
Ramification exponent $e$: $1$
Residue field degree $f$: $3$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{103}$
Root number: $1$
$|\Gal(K/\Q_{ 103 })|$: $3$
This field is Galois and abelian over $\Q_{103}.$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 103 }$.

Unramified/totally ramified tower

Unramified subfield:103.3.0.1 $\cong \Q_{103}(t)$ where $t$ is a root of \( x^{3} - x + 29 \)  Toggle raw display
Relative Eisenstein polynomial:\( x - 103 \)$\ \in\Q_{103}(t)[x]$  Toggle raw display

Invariants of the Galois closure

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