# Manual Unique factorization domains

A unique factorization domain, called UFD for short, is any integral domain in which every nonzero noninvertible element has a unique factorization, i.e.

## Unique factorization domain - Example Problems

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Cancel Submit. Your feedback will be reviewed. Add a definition. Since a unique factorization domain is precisely a ring in which an analogue of the fundamental theorem of arithmetic holds, this question is readily answered. From Wikipedia. This ring is a frequently cited example of an integral domain that is not a unique factorization domain.

As for every principal ideal domain, is also a unique factorization domain. A unique factorization domain is not necessarily a noetherian ring. Thus, a number field has class number 1 if and only if its ring of integers is a principal ideal domain and thus a unique factorization domain. The greatest common divisor may be defined and exists, more generally, for multivariate polynomials over a field or the ring of integers, and also over a unique factorization domain.

This property of unique factorization is commonly expressed by saying that the polynomial rings over a field or a unique factorization domain are unique factorization domains. A third reason is that the theory and the algorithms for the multivariate case and for coefficients in a unique factorization domain are strongly based on this particular case. The rings where every irreducible is prime are called unique factorization domains. The converse does not hold in general, but does hold for unique factorization domains. The fundamental theorem of arithmetic continues to hold in unique factorization domains.

This led to the study of unique factorization domains, which generalize what was just illustrated in the integers. They showed that regular local rings of dimension 3 are unique factorization domains, and had previously shown that this implies that all regular local rings are unique factorization domains.

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BETA Add a definition. Part of speech Choose noun, verb, etc. Since every prime ideal has finite height, it contains height one prime ideal induction on height which is principal. By 2 , the ring is a UFD. In mathematics, a unique factorization domain UFD is an integral domain a non-zero commutative ring in which the product of non-zero elements is non-zero in which every non-zero non-unit element can be written as a product of prime elements or irreducible elements , uniquely up to order and units, analogous to the fundamental theorem of arithmetic for the integers.

UFDs are sometimes called factorial rings, following the terminology of Bourbaki. In mathematics, factorization or factorisation, see English spelling differences or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. However, a meaningful factorization for a rational number or a rational function can be obtained by writing it in lowest terms and separately factoring its numerator and denominator.

Factorization was first considered by ancient Greek mathematicians in the. In algebra, Gauss's lemma, named after Carl Friedrich Gauss, is a statement about polynomials over the integers, or, more generally, over a unique factorization domain that is, a ring that has a unique factorization property similar to the fundamental theorem of arithmetic. Gauss's lemma underlies all the theory of factorization and greatest common divisors of such polynomials.

Gauss's lemma asserts that the product of two primitive polynomials is primitive a polynomial with integer coefficients is primitive if it has 1 as a greatest common divisor of its coefficients. A corollary of Gauss's lemma, sometimes also called Gauss's lemma, is that a primitive polynomial is irreducible over the integers if and only if it is irreducible over the rational numbers. More generally, a primitive polynomial has the same complete factorization over the integers and over the rational numbers. In the case of coefficients in a unique factorization domain R, "rational numbers" must be replaced by "field of fractions o.

In mathematics, the noncommutative unique factorization domain is the noncommutative counterpart of the commutative or classical unique factorization domain UFD. Example The ring of integral quaternions. References R. The unique factorization theorem was proved by Gauss with his book Disquisitiones Arithmeticae. The theorem says two things for this example: first, that can be represented as a product of primes, and second, that no matter how this is done, there will always be exactly four 2s, one 3, two 5s, and no other primes in the product. The requirement that the factors be prime is necessary: factorizations containing composite numbers may.

In mathematics, an irreducible polynomial or prime polynomial is, roughly speaking, a non-constant polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the field or ring to which the coefficients of the polynomial and its possible factors are supposed to belong. A polynomial that is irreducible over any field containing the coefficients is absolutely irred.

In algebra, the content of a polynomial with integer coefficients or, more generally, with coefficients in a unique factorization domain is the greatest common divisor of its coefficients. The primitive part of such a polynomial is the quotient of the polynomial by its content. Thus a polynomial is the product of its primitive part and its content, and this factorization is unique up to the multiplication of the content by a unit of the ring of the coefficients and the multiplication of the primitive part by the inverse of the unit.

A polynomial is primitive if its content equals 1. Thus the primitive part of a polynomial is a primitive polynomial. Gauss's lemma for polynomials states that the product of primitive polynomials with coefficients in the same unique factorization domain also is primitive.

This implies that the content and the primitive part of the product of two polynomials are, respectively, the product of the contents and the product of the primitive parts. As the computation of greate.

Unique factorization and its difficulties I Data Structures in Mathematics Math Foundations 198

Equivalently, any two elements of R have a least common multiple LCM. Among the GCD domains,. In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.

More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors e. The distinction is that a principal ideal ring may have zero divisors whereas a principal ideal domain cannot.

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4. Principal ideal domains are thus mathematical objects that behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition into prime elements so an analogue of the fundamental theorem of arithmetic holds ; any two elements of a PID have a greatest common divisor although it may not be possible to find it using the Euclidean algorithm.

Principal ideal domains are noetherian, they are integrally closed, the. In mathematics, more specifically in ring theory, a Euclidean domain also called a Euclidean ring is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of the integers. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements.

Also every ideal in a Euclidean domain is principal, which implies a suitable generalization of the fundamental theorem of arithmetic: every Euclidean domain is a unique factorization domain. It is important to compare the class of Euclidean domains with the larger class of principal ideal domains PIDs.

An arbitrary PID has much the same "structural properties" of a Euclidea. In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. It does not have a total ordering that respects arithmetic. Since the Gaussian integers are closed under addition and multiplication, they form a commutative ring, which is a subring of the field of complex numbers.

It is thus an integral domain. When considered within the complex plane,. In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals.

### Principal ideal domains

It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below. A field is a commutative ring in which there are no nontrivial proper ideals, so that any field is a Dedekind domain, however in a rather vacuous way. Some authors add the requirement that a Dedekind domain not be a field.

Many more authors state theorems for Dedekind domains with the implicit proviso that they may require trivial modifications for the case of fields. An immediate consequence of the definition is that every principal ideal domain PID is a Dedekind domain.

## unique-factorization-domain

The prehistory of Dedekind domains In. In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. This article follows the convention that rings have a multiplicative identity, generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity.

Some sources, notably Lang, use the term entire ring.

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In commutative algebra, an integrally closed domain A is an integral domain whose integral closure in its field of fractions is A itself. Many well-studied domains are integrally closed: Fields, the ring of integers Z, unique factorization domains and regular local rings are all integrally closed. In other words, A is not integrally closed. The prime numbers are the natural numbers greater than one that are not products of two smaller natural numbers.