First-degree prime ideals of composite extensions

Abstract

Let Q(α){\mathbb{Q}}\left(\alpha ) and Q(β){\mathbb{Q}}\left(\beta ) be linearly disjoint number fields and let Q(θ){\mathbb{Q}}\left(\theta ) be their compositum. We prove that the first-degree prime ideals (FDPIs) of Z[θ]{\mathbb{Z}}\left[\theta ] may almost always be constructed in terms of the FDPIs of Z[α]{\mathbb{Z}}\left[\alpha ] and Z[β]{\mathbb{Z}}\left[\beta ], 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 d,e∈Zd,e\in {\mathbb{Z}}, such a correspondence almost always respects the divisibility of principal ideals of the form (e+dθ)Z[θ]\left(e+d\theta ){\mathbb{Z}}\left[\theta ], 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.

Keywords

• first-degree prime ideals
• principal ideal factorization
• linearly disjoint extensions
• 11y05
• 11y40
• 12f05