First-degree prime ideals of composite extensions

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="double-struck">Q</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {Q}(α ) and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="double-struck">Q</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>β</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {Q}(β ) be linearly disjoint number fields and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="double-struck">Q</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>θ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {Q}(θ ) be their compositum. We prove that the first-degree prime ideals (FDPIs) of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="double-struck">Z</m:mi> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mi>θ</m:mi> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:math> {Z}[θ ] may almost always be constructed in terms of the FDPIs of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="double-struck">Z</m:mi> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:math> {Z}[α ] and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="double-struck">Z</m:mi> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mi>β</m:mi> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:math> {Z}[β ] , and vice versa . We identify the cases where this correspondence does not hold, and provide explicit counterexamples for each obstruction. We show that for every pair of coprime integers <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>d</m:mi> <m:mo>,</m:mo> <m:mi>e</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">Z</m:mi> </m:math> d,e∈ {Z} , such a correspondence almost always respects the divisibility of principal ideals of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>e</m:mi> <m:mo>+</m:mo> <m:mi>d</m:mi> <m:mi>θ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi mathvariant="double-struck">Z</m:mi> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mi>θ</m:mi> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:math> (e+dθ ){Z}[θ ] , with a few exceptions that we characterize. Finally, we establish the asymptotic computational improvement of such an approach, and we verify the reduction in time needed for computing such primes for certain concrete cases.