Properties

Label 59.7.0.1
Base \(\Q_{59}\)
Degree \(7\)
e \(1\)
f \(7\)
c \(0\)
Galois group $C_7$ (as 7T1)

Related objects

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Defining polynomial

\( x^{7} - x + 16 \)

Invariants

Base field: $\Q_{59}$
Degree $d$: $7$
Ramification exponent $e$: $1$
Residue field degree $f$: $7$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{59}$
Root number: $1$
$|\Gal(K/\Q_{ 59 })|$: $7$
This field is Galois and abelian over $\Q_{59}.$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 59 }$.

Unramified/totally ramified tower

Unramified subfield:59.7.0.1 $\cong \Q_{59}(t)$ where $t$ is a root of \( x^{7} - x + 16 \)
Relative Eisenstein polynomial:$ x - 59 \in\Q_{59}(t)[x]$

Invariants of the Galois closure

Galois group:$C_7$ (as 7T1)
Inertia group:trivial
Unramified degree:$7$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{7} - x^{6} - 18 x^{5} + 35 x^{4} + 38 x^{3} - 104 x^{2} + 7 x + 49$