2022/11/24 by Singhal, Deepesh, Lin, Yuxin · 1 citation
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2211.13822
Denote the set of algebraic numbers as ℚ and the set of algebraic integers as ℤ. For γ∈ℚ, consider its irreducible polynomial in ℤ[x], Fγ(x)=anxn+…+a0. Denote e(γ)=gcd(an,an-1,…,a1). Drungilas, Dubickas and Jankauskas show in a recent paper that ℤ[γ]∩ ℚ=\α∈ℚ| \p| vp(α)<0\⊆ \p| p|e(γ)\\. Given a number field K and γ∈ℚ, we show that there is a subset X(K,γ)⊆ Spec(OK), for which OK[γ]∩ K=\α∈ K| \\mathfrakp| v_\mathfrakp(α)<0\⊆ X(K,γ)\. We prove that OK[γ]∩ K is a principal ideal domain if and only if the primes in X(K,γ) generate the class group of OK. We show that given γ∈ ℚ, we can find a finite set S⊆ ℤ, such that for every number field K, we have X(K,γ)=\\mathfrakp\inSpec(OK)| \mathfrakp∩ S≠ ∅\. We study how this set S relates to the ring ℤ[γ] and the ideal \mathfrakDγ=\a∈ℤ| aγ∈ℤ\ of ℤ. We also show that γ1,γ2∈ ℚ satisfy \mathfrakDγ1=\mathfrakDγ2 if and only if X(K,γ1)=X(K,γ2) for all number fields K.