Properties

Label 31.5.4.5
Base \(\Q_{31}\)
Degree \(5\)
e \(5\)
f \(1\)
c \(4\)
Galois group $C_5$ (as 5T1)

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Defining polynomial

\(x^{5} + 248\) Copy content Toggle raw display

Invariants

Base field: $\Q_{31}$
Degree $d$: $5$
Ramification exponent $e$: $5$
Residue field degree $f$: $1$
Discriminant exponent $c$: $4$
Discriminant root field: $\Q_{31}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 31 }) }$: $5$
This field is Galois and abelian over $\Q_{31}.$
Visible slopes:None

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 31 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{31}$
Relative Eisenstein polynomial: \( x^{5} + 248 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{4} + 5z^{3} + 10z^{2} + 10z + 5$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_5$ (as 5T1)
Inertia group:$C_5$ (as 5T1)
Wild inertia group:$C_1$
Unramified degree:$1$
Tame degree:$5$
Wild slopes:None
Galois mean slope:$4/5$
Galois splitting model:$x^{5} - x^{4} - 136 x^{3} - 723 x^{2} - 1053 x - 67$