Earlier we justified our proof of the fundamental theorem of arithmetic by arguing that the result was not obvious, since its analogue in the ring E of even integers did not hold. Here we will explore a far more exciting explanation: essentially the same proof can be used to show that certain other “integer-like rings” have the unique factorization property. The unique factorization property ca...

1 Rings Definition 1.1 (Rings). A ring R is a set with two binary operations + and · called addition and multiplication respectively such that: (1) (R, +) is an Abelian group with identity element 0. (2) (R, ·) is a monoid with identity element 1. (3) Multiplication distributes over addition: a · (b + c) = a · b + a · c and (b + c) · a = b · a + c · a for all a, b, c ∈ R. A ring is called commu...

In this paper we study the algorithmic problem of nding the ring of integers of a given algebraic number eld. In practice, this problem is often considered to be well-solved, but theoretical results indicate that it is intractable for number elds that are deened by equations with very large coeecients. Such elds occur in the number eld sieve algorithm for factoring integers. Applying a variant ...

In this paper we study the algorithmic problem of finding the ring of integers of a given algebraic number field. In practice, this problem is often considered to be well-solved, but theoretical results indicate that it is intractable for number fields that are defined by equations with very large coefficients. Such fields occur in the number field sieve algorithm for factoring integers. Applyi...

Let G be a finite group and k a fixed algebraically closed field of characteristic p>0 . If p is odd, let H(; be the subring of H*(G,k) consisting of elements of even degree; following [20-22] we take H~=H*(G,k) if p=2, though one could just as well use the subring of elements of even degree for all p. H a is a finitely generated commutative k-algebra [13], and we let Va denote its associated a...

Let (K,v) be a valued field, and μ an inductive valuation on K[x] extending v. Gμ the graded algebra of over K[x], κ maximal subfield subring formed by homogeneous elements degree zero. In this paper, we find algorithm to compute field residual polynomial operator Rμ:K[x]→κ[y], where y is another indeterminate, without any need perform computations in algebra. This leads OM factorization separa...

In this paper, we give a new method for constructing LCD codes. We employ group rings and well known map that sends ring elements to subring of the n×n matrices obtain Our construction guarantees our codes are also codes, namely, ideals in ring. show with certain condition on element v, one can construct non-trivial Moreover, by adding more constraints reversible. present many examples binary w...

For a moduli space $${\mathsf M}$$ of stable sheaves over K3 surface X, we propose series conjectural identities in the Chow rings $$CH_\star ({\mathsf M}\times X^\ell ),\, \ell \ge 1,$$ generalizing classic Beauville–Voisin identity for surface. We emphasize consequences conjecture structure tautological subring $$R_\star M}) \subset CH_\star M}).$$ The places all classes lowest piece natural ...