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Class Groups
Published in Richard A. Mollin, Algebraic Number Theory Second, 2011
In §3.5, we will establish the celebrated Dirichlet unit theorem. We set the stage in this section by establishing results on the finite component of the unit group, namely the group of roots of unity. Of fundamental importance is the ring of integers of a cyclotomic field. This will become even more transparent later when we establish the Kronecker-Weber Theorem. First however, we need the following crucial result on a compositum of fields due to Hilbert-see Biography 3.4 on page 94. The reader should therefore be familiar with the discussion surrounding Application A.1 on page 325.
Paradigm crisis in the step from tertiary to secondary mathematics education
Published in International Journal of Mathematical Education in Science and Technology, 2022
Indeed, when the disciplinary didactic paradigm shared in the IPM is compared with the PMM, one finds a consequence of deductivism, directly conflicting with the PMM: the oblivion of the problems at the basis of the theories and how those problems contributed to shape those theories. Hence, in the current didactic paradigm of the IPM, the axioms and definitions at the beginning of the theories appear out of the blue, as pure syntactic facts without any reference to an interpretation in terms of a system which truly motivates our actions and considerations. For instance, in the case of the set-theoretical definition of the product in the ring of rational numbers as (a/b)·(c/d) = (a·c)/(b·d), in the IPM one typically studies the compatibility with the set-theoretical definition of the product in the ring of integers, but one does not study any meaning, any interpretation of the product, any justification of the definition in terms of an underlying system (Gascón & Nicolás, 2018a).
A Component-Position Model, Analysis and Design for Order-of-Addition Experiments
Published in Technometrics, 2021
Jian-Feng Yang, Fasheng Sun, Hongquan Xu
We need the concept of Galois fields in order to describe our next construction method for component orthogonal arrays. A Galois field (or finite field) is a field that contains a finite number of elements, on which the operations of multiplication, addition, subtraction and division are defined and satisfy the rules of arithmetic. The number of elements, called the order of a Galois field, must be a prime power. For any prime p and positive integer u, there is a unique Galois field of order pu up to isomorphism. Let be a Galois field. For a prime number p, is simply the ring of integers modulo p. For u > 1, . In other words, the elements of are polynomials with degree less than u and coefficients from GF(p). The addition of is ordinary polynomial addition with coefficients modulo p, and the multiplication is ordinary polynomial multiplication and then modulo a given irreducible polynomial of degree u. An irreducible polynomial (or primitive polynomial) is a polynomial that cannot be factored into the product of two non-constant polynomials. For a selected primitive polynomial over GF(p), see Table A.19 of Hedayat et al. (1999).
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: .