Nonunique factorization and principalization in number fields

Non-unique Factorization and Principalization in Number Fields

Following what is basically Kummer’s relatively neglected approach to non-unique factorization, we determine the structure of the irreducible factorizations of an element n in the ring of integers of a number field K. Consequently, we give a combinatorial expression for the number of irreducible factorizations of n in the ring. When K is quadratic, we show in certain cases how quadratic forms c...

Polynomial factorization algorithms over number fields

Factorization algorithms over Q[X] and Fp[X] are key tools of computational number theory. Many algorithms over number fields rely on the possibility of factoring polynomials in those fields. Because of the recent development of relative methods in computational number theory, see for example (Cohen et al. 1998, Daberkow and Pohst 1995), efficient generalizations of factorization algorithms to ...

Applications of Prime Factorization of Ideals in Number Fields

For a number fieldK, that is, a finite extension of Q, and a prime number p, a fundamental theorem of algebraic number theory implies that the ideal (p) ⊆ OK factors uniquely into prime ideals as (p) = p1 1 · · · p eg g . In this paper we explore different interpretations of this using the factorization of polynomials in finite and p-adic fields and Galois theory. In particular, we present some...

Uniqueness of Factorization in Quadratic Fields

Monomial Principalization in the Singular Setting

We generalize an algorithm by Goward for principalization of monomial ideals in nonsingular varieties to work on any scheme of finite type over a field. The normal crossings condition considered by Goward is weakened to the condition that components of the generating divisors meet as complete intersections.