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Integral Domains, Ideals, and Unique Factorization
Published in Richard A. Mollin, Algebraic Number Theory Second, 2011
If D is an integral domain in which every nonzero, nonunit can be represented as a finite product of irreducible elements of D, then D is called a factorization domain. A factorization domain in which any nonzero, nonunit can be expressed as a product of irreducibles that is unique up to units and the order of the factors is called a unique factorization domain (UFD)
Arc length of function graphs via Taylor's formula
Published in International Journal of Mathematical Education in Science and Technology, 2021
Suppose that we seek two functions p and q such that and where g is some function to which we can find a primitive function G. This implies that or equivalently that . One way to accomplish this is if for some function r of reasonably simple type. This means that is a Pythagorean triple of functions. It is a classical result in number theory that such triples, consisting of integers, can be parametrized by , and , where k, m and n are positive integers with m>n, and with m and n coprime and not both odd (see e.g. Long, 1972). In Kubota (1972) Kubota has shown that the same kind of result holds in any unique factorization domain (UFD). In particular, it holds for polynomial rings , since they are Euclidean domains and hence UFD's. The bottom line is that we can use this kind of parametrization to yield examples of rectifiable curves in the following way.
Divisibility tests for polynomials
Published in International Journal of Mathematical Education in Science and Technology, 2020
In other words, a domain R is called unique factorization domain iff every element neither 0 nor a unit element has a factorization into irreducible elements of R, and this factorization is unique up to order and units. It is easy to prove that , the ring of integers, is a UFD. Indeed, the statement: ‘ is a UFD’ is just a restatement of the fundamental theorem of arithmetic. Using more complex arguments, it also can be proved that the domain is a UFD (see Rotman, 2003, Example 11.47, p. 926). On the contrary, the ring is not a UFD. In fact 6 can be written as a product of irreducibles in two different ways: .