Properties

Label 31.10.9.3
Base \(\Q_{31}\)
Degree \(10\)
e \(10\)
f \(1\)
c \(9\)
Galois group $C_{10}$ (as 10T1)

Related objects

Downloads

Learn more

Defining polynomial

\(x^{10} + 527\) Copy content Toggle raw display

Invariants

Base field: $\Q_{31}$
Degree $d$: $10$
Ramification exponent $e$: $10$
Residue field degree $f$: $1$
Discriminant exponent $c$: $9$
Discriminant root field: $\Q_{31}(\sqrt{31})$
Root number: $-i$
$\card{ \Gal(K/\Q_{ 31 }) }$: $10$
This field is Galois and abelian over $\Q_{31}.$
Visible slopes:None

Intermediate fields

$\Q_{31}(\sqrt{31})$, 31.5.4.5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:$\Q_{31}$
Relative Eisenstein polynomial: \( x^{10} + 527 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{9} + 10z^{8} + 14z^{7} + 27z^{6} + 24z^{5} + 4z^{4} + 24z^{3} + 27z^{2} + 14z + 10$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_{10}$ (as 10T1)
Inertia group:$C_{10}$ (as 10T1)
Wild inertia group:$C_1$
Unramified degree:$1$
Tame degree:$10$
Wild slopes:None
Galois mean slope:$9/10$
Galois splitting model:Not computed