Primes in denominators of algebraic numbers.

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 ] , F γ (x) = a n x n + ⋯ + a 0 . Denote e (γ) = gcd (a n , a n − 1 , ... , a 1). Drungilas, Dubickas and Jankauskas show in a recent paper that ℤ [ γ ] ∩ ℚ = { α ∈ ℚ | { p | v p (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 ( K) , for which K [ γ ] ∩ K = { α ∈ K | { | v (α) < 0 } ⊆ X (K , γ) }. We prove that K [ γ ] ∩ K is a principal ideal domain if and only if the primes in X (K , γ) generate the class group of K . 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) = { ∈ Spec ( K) | p ∩ S ≠ ∅ }. We study how this set S relates to the ring ℤ ¯ [ γ ] and the ideal γ = { a ∈ ℤ ¯ | a γ ∈ ℤ ¯ } of ℤ ¯. We also show that γ 1 , γ 2 ∈ ℚ ¯ satisfy γ 1 = γ 2 if and only if X (K , γ 1) = X (K , γ 2) for all number fields K. [ABSTRACT FROM AUTHOR]