An introductory course in commutative algebra

Let q be a power of a prime p and Fq be the finite field with q elements. Denote by Fq[X] and Fq(X) the ring of polynomials with coefficients in Fq and the quotient field of Fq[X] respectively. For each P/Q ∈ Fq(X) define |P/Q| = qdeg(P )−deg(Q) where for an element g ∈ Fp[X]we have denoted its degree by deg(g). Note that with respect to the valuation function | · | the integral domain Fq[X] is a Euclidean domain and so also a principal ideal domain and a unique factorization domain. In particular Bezout’s identity is satisfied. Let Fq((X−1)) denote the field of formal Laurent series Fq((X)) = {anX + · · · + a0 + a−1X + · · · : n ∈ Z, ai ∈ Fq},

This thesis deals with two properties of arithmetic functions, namely, factorization and independence. In 1984, Rearick pointed out that the ring of arithmetic functions is a unique factorization domain but is not a principal ideal domain and so is not a Euclidean domain. Without the Euclidean algorithm, the problem of factorizing arithmetic functions becomes quite difficult. In the first part of this thesis, we propose a technique of factorizing certain classes of arithmetic functions and prove some results about factorization which are based mainly on the norms of such functions. The technique is a generalization of the original works of Rearick which consists of solving a special system of linear differential equations whose coefficients are polynomials in the function to be factorized. The solutions of this system are proved to be the sought after factors with increasing norms. Examples illustrating the technique are also given. In 1986, Shapiro and Sparer made an extensive study of algebraic independence of Dirichlet series. Since the ring of Dirichlet series is isomorphic to that of arithmetic functions, the study in one setting is then equivalent to the other. Shapiro and Sparer's investigation began with a theorem asserting that Dirichlet series are algebraically independent if their Jacobian does not vanish, which is classical in the case of real-valued functions. Taking the Riemann zeta function as a building block, they discovered that Dirichlet series algebraically dependent on the zeta function can uniquely be represented as power series in the logarithms of zeta function. A number of algebraic dependence results of Dirichlet series were derived as consequences. Shapiro and Sparer then went on to investigate analogous results for formal generalized Dirichlet series. Results in the second part of this thesis either extend or simplify some of Shapiro and Sperer's results. These include, for example, replacing the zeta function by Dirichlet series with completely multiplicative coefficients to obtain similar log-series expansions, dependence of series with infinite support, and dependence of non-units whose norms are relatively prime.

Euclidean domain is also known as Euclidean ring. It is an integral domain which is associated with at least one Euclidean function [1]. A non zero commutative ring in which of product two non zero element is non zero is known as integral domain. Integral domain is generalization of ring of integers [2]. In Euclidean domain we use Euclidean algorithm for calculating greatest common divisors of two elements. Greatest common divisors of two elements always exist and this can be represented in linear combination of these elements. Euclidean domain has ideals and every ideal of Euclidean domain is principal ideal [3]. Euclidean domain has property of unique factorization domain and every Euclidean domain is a unique factorization domain it means every non zero non unit element can be written as product of prime elements. Key words: algorithm, ideal, trichotomy, polynomials

Let D be a principal ideal domain (PID), I be an ideal of D, and X be an indeterminate over D. Let [D;I][X] be the subring of the power series ring consisting of all power series in such that for all large i. By definition, the polynomial ring and the power series ring are special cases of when and I = D, respectively. In this article, we investigate the ring in the case I is a nonzero proper ideal of D. We prove that R is a two-dimensional non-Noetherian ring. For each maximal ideal P of D, it is shown that is a height-one prime ideal of R. The set of units of R is given and the spectrum of R is also described. Unlike the power series ring the ring R is not a unique factorization domain (UFD). Furthermore, when I is a nonzero prime ideal, R does not satisfy both ACCP and the atomic property. In obtaining results on R, we introduce and sometimes use results on the ring RS , where I = dD with and Closely related to R, the ring RS is shown to be a Noetherian UFD with Krull dimension at most two. Moreover, RS has Krull dimension two exactly when I is not contained in the Jacobson radical of D; otherwise RS is a PID.

Let A be a domain, that is, a commutative ring with unit element (different from 0), having no zero-divisors (except 0). Let K be its field of quotients.

We investigate the problem on the validity of the Lazard theorem analogue (or the Kronecker–Vahlen theorem), which states that the least remainder Euclidean Algorithm (EA) has the shortest length over any other versions of EA, in unique factorization domains. There is obtained the existence criterion to represent a fixed element of the fractions field of a unique factorization domain in the form of a continued fraction of a fixed length. This criterion enables to obtain a formula for the length of the least remainder (on norm) EA as a function of elements, with respect to which EA is applied. This result gives us the class T of unique factorization domains, for which the Lazard theorem analogue is valid. We propose algorithms to check whether the given unique factorization domain belongs to the class T. We find the necessary and sufficient conditions under which the number of steps in the worst case of the least remainder EA grows not faster than logarithm. In particular, these results hold for the least remainder EA in any Euclidean quadratic domain. We provide counterexamples, which show the essentiality of the conditions in the obtained theorems.

This paper describes the implementation of recursive algorithms in unique factorization domains, namely multivariate polynomial greatest common divisors (gcd) and factorization into irreducible parts in the Java computer algebra library (JAS). The implementation of gcds, resultants and factorization is part of the essential building blocks for any computation in algebraic geometry, in particular in automated deduction in geometry. There are various implementations of these algorithms in procedural programming languages. Our aim is an implementation in a modern object oriented programming language with generic data types, as it is provided by Java programming language. We exemplify that the type design and implementation of JAS is suitable for the implementation of several greatest common divisor algorithms and factorization of multivariate polynomials. Due to the design we can employ this package in very general settings not commonly seen in other computer algebra systems. As for example, in the coefficient arithmetic for advanced Grobner basis computations like in polynomial rings over rational function fields or (finite, commutative) regular rings. The new package provides factory methods for the selection of one of the several implementations for non experts. Further we introduce a parallel proxy for gcd implementations which runs different implementations concurrently.

Integral domains are commutative rings whose non-zero elements are closed under multiplication. If each nonzero element is a unit, the domain is called a field and is shipped off to Chap. 11. For the domains D which remain, divisibility is a central question. A prime ideal has the property that elements outside the ideal are closed under multiplication. A non-zero element \(a\in D\) is said to be prime if the principle ideal Da which it generates is a prime ideal. D is a unique factorization domain (or UFD) if any expression of an element as a product of prime elements is unique up to the order of the factors and the replacement of any prime factor by a unit multiple. If D is a UFD, so is the polynomial ring D[X] where X is a finite set of commuting indeterminates. In some cases, the unique factorization property can be determined by the localizations of a domain. Euclidean domains (like the integers, Gaussian and Eisenstein numbers) are UFD’s, but many domains are not. One enormous class of domains (which includes the algebraic integers) is obtained the following way: Suppose K a field which is finite-dimensional over a subfield F which, in turn, is the field of fractions of an integral domain D. One can then define the ring \(\mathcal{O}_D(K)\) of elements of K which are integral with respect to D. Under modest conditions, the integral domain \(\mathcal{O}_D(K)\), will become a Noetherian domain in which every prime ideal is maximal—a so-called Dedekind domain. Although not UFD’s, Dedekind domains offer a door prize: every ideal can be uniquely expressed as a product of prime ideals (up to the order of the factors, of course).

The fundamental theorem of algebra states that every non-constant polynomial with complex coefficients can be factorized on linear factors with coefficients from the same field. However, this is not true for numerous other equally important and interesting fields such as, for example, finite fields. All existing finite fields are of the q=pk order, where p is a prime number and k is a positive integer. Any two finite fields of the same order are mutually isomorfic, therefore the field of q order is unique; it is also known as »the Galois field«, named after the French mathematician Evariste Galois (1811-1832). This thesis deals with factorization of polynomials over finite fields. The general rule states that some polynomials are not reducible, yet each polynomial with coefficients of the field can be uniquely factorized into irreducible factors. There are various known algorithms that can determine whether or not a polynomial is reducible and – in the case of reducible polynomials – also find this factorization. There are several such algorithms, however, this thesis shall focus on the Berlekamp's algorithm. The introductory chapter introduces the historical development of the various approaches to polynomial factorization and presents some modern applications that are currently in use. The second chapter focuses on basic concepts and characteristics of algebraic structures and some classic results that are crucial for the understanding of the algorithm (e.g. groups, rings, fields, division rings and polynomial rings.). The algorithm also uses the Chinese remainder theorem for polynomials and Euclidean algorithm for calculating the greatest common divisor of two polynomials. The first part of Chapter 3 (i.e. the main chapter of the thesis) demonstrates 4 naive methods for the finding of polynomial factorization. For each of the methods we first introduced factorization of a polynomial over real numbers and then continued with an example of a polynomial with coefficients in the finite field. The sieve method, for instance, builds a list of irreducible polynomials of low degrees. Irreducible polynomials up to (and including) degree 6 over the finite fields Z2, Z3, Z5 and Z7 can be calculated using our own code made in the VisualC# programming language. These naive methods are followed by a detailed description of the Berlekamp's algorithm with theorems, proofs and various examples. At the end of section we present our implementation of the algorithm in the computational algebra system MAGMA.

In this paper, we extend the concept of lcem matrices beyond the classical domain of natural integers into the domain of unique factorization domains. We investigate the structure of these matrix types when applied to both arbitrary sets and gced-closed sets. Furthermore, we find the determinant, the trace and the inverse of such matrices. To simplify these ideas, we employ domains such as the Gaussian integers domain and the domain of polynomials defined over finite fields.

Tame and wild coordinates of [formula omitted

Given a certain factorization property of a ring [Formula: see text], we can ask if this property extends to the polynomial ring over [Formula: see text] or vice versa. For example, it is well known that [Formula: see text] is a unique factorization domain if and only if [Formula: see text] is a unique factorization domain. If [Formula: see text] is not a domain, this is no longer true. In this paper, we survey unique factorization in commutative rings with zero divisors, and characterize when a polynomial ring over an arbitrary commutative ring has unique factorization.

The study of factorization in integral domains has played an important role in commutative algebra for many years. Much of this work has con­centrated on the study of unique factorization domains (UFDs). Beyond the realm of UFDs, there is a large class of integral domains for which each nonunit can be factored as a product of irreducibles, yet the factorization may not be unique. Classically, Dedekind domains are such domains. A Dedekind domain D is a UFD if and only if its ideal class group is trivial. Hence the size of the class group becomes a measure of how far the domain D is from having “unique factorization”. The concept of the class group and its relation to factorization properties extends also to the more general class of Krull domains. It is within the context of these domains that we consider in this paper general questions concerning the lengths of factorization in the non-UFD setting.

An integral domain D is called a unique factorization domain (UFD) if the following conditions are satisfied: (1) Every element of D that is neither O nor a unit can be factored into a product of a finite number of irreducibles, and (2) if p1, … , pr and q1 …, qs are two factorizations of the same element of D into irreducibles, then r = s and qj can be renumbered so that pi and qi are associates. It has been known the following equivalent conditions: Let D be an integral domain, the following are equivalent: (i) D is a UFD, (ii) D is a GCD domain satisfying the ascending chain condition on principal ideals, and (iii) D satisfies the ascending chain condition on principal ideals and every irreducible element of D is a prime element of D. The concept and the equivalent conditions on UFD, motivate some studies that may apply the concept of factorization to modules in order to obtain a definition of a unique factorization module (UFM). First, the concept of irreducible elements is given in the module which will play an important role in defining the UFM. The definition of primitive elements, pure submodules, least common multiple and greatest common divisor in a module is also given. Next, the definition and characterization of a UFM will be presented. The results of this study is providing the sufficient and necessary conditions of a module to be UFM.

We consider a question raised by Mowaffaq Hajja about the structure of a principal ideal domain R having the property that 1 is the only unit of R. We also examine this unit condition for the case where R is a Euclidean domain. We prove that a finitely generated Euclidean domain having 1 as its only unit is isomorphic to the field with two elements F 2 or to the polynomial ring F 2[X]. On the other hand, we establish existence of finitely generated principal ideal domains R such that 1 is the only unit of R and R is not isomorphic to F 2 or to F 2[X]. We also construct principal ideal domains R of infinite transcendence degree over F 2 with the property that 1 is the only unit of R. *Part of this work was prepared while M. Roitman enjoyed the hospitality of Purdue University.