$p$-rationality

Hecke provides predicates for testing whether an absolute simple number field is quasi-$p$-rational or $p$-rational for a given prime $p$.

Quasi-$p$-rationality

is_quasi_p_rational — Function

is_quasi_p_rational(K::AbsSimpleNumField, p; GRH::Bool = false)

Return whether the number field $K$ is quasi-$p$-rational.

$p$-rationality

is_p_rational — Function

is_p_rational(K::AbsSimpleNumField, p; GRH::Bool = false)

Return whether the number field $K$ is $p$-rational at $p$.

See also is_quasi_p_rational and is_real_cyclotomic_field_p_rational for an improved version that works for real cyclotomic fields and does not require GRH.

Examples

julia> K, = cyclotomic_real_subfield(15);

julia> is_p_rational(K, 13)
false

source

Real cyclotomic fields

For maximal real subfields of cyclotomic fields, a specialized predicate is available.

is_real_cyclotomic_field_p_rational — Function

is_real_cyclotomic_field_p_rational(n::Int, p; GRH::Bool = false)

Return whether the maximal real subfield of the cyclotomic field of conductor $n$ is $p$-rational at $p$. The conductor $n$ must not be congruent to $2$ modulo $4$.

See also is_quasi_p_rational and is_p_rational for versions that work for any number field.

Examples

julia> is_real_cyclotomic_field_p_rational(15, 13)
false

julia> p = ZZ(2)^127 - 1;

julia> is_real_cyclotomic_field_p_rational(5, p)
true

source