Abstract
In this paper, we give an elementary proof of the fact that the rings are unique factorization domains for the values d = 3, 7, 11, 19, 43, 67, 163. While the result in itself is well known, our proof is new and completely elementary and uses neither the Minkowski convex body theorem, nor the Dedekind and Hasse theorems. Furthermore, it does not use either the theory of algebraic integers, or the theory of Noetherian rings. It only uses basic notions from the theory of commutative rings.
Highlights
The main purpose of this paper is give an elementary proof of the fact that the rings Z1+ −d 2 are unique factorization domains, for the values d =3, 7, 11, 19, 43, 67, 163
In all what follows α ∈ C is a root of the irreducible polynomial x2 +tx+q ∈ Z[x]
If π ∈ Z[α] is such that N (π) is prime number, π is prime in Z[α]
Summary
The main purpose of this paper is give an elementary proof of the fact that the rings Z. 315), our proof is new and completely elementary and uses neither the Minkowski convex body theorem (See [2] Chapter VIII), nor the Dedekind and Hasse theorems It does not use either the theory of algebraic integers, or the theory of Noetherian rings. It only uses basic notions from the theory of commutative rings
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