Factorization of monic polynomials

We prove a uniqueness result about the factorization of a monic polynomial over a general commutative ring into comaximal factors. We apply this result to address several questions raised by Steve McAdam. These questions, inspired by Hensel's Lemma, concern properties of prime ideals and the factoring of monic polynomials modulo prime ideals.

     

The Drazin Inverse of a Morphism with a Sequence of (epic, monic) Factorizations

§ 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　Ｉｎ [1 ],ｔｈｅａｕｔｈｏｒｄｅｆｉｎｅｄａｃｃｕｒａｔｅｌｙｔｈｅＤｒａｚｉｎｉｎｖｅｒｓｅｏｆａｍｏｒｐｈｉｓｍ .Ｐｒｅｖｉｏｕｓｌｙ ,ｓｏｍｅｎｅｃｅｓｓａｒｙａｎｄｓｕｆｆｉｃｉｅｎｔｃｏｎｄｉｔｉｏｎｓｗｅｒｅｏｂｔａｉｎｅｄｆｏｒａｍｏｒｐｈｉｓｍｗｉｔｈ (ｅｐｉｃ ,ｍｏｎｉｃ)ｆａｃｔｏｒｉｚａｔｉｏｎｔｏｈａｖｅｔｈｅｇｒｏｕｐｉｎｖｅｒｓｅ (ｓｅｅ [2 ],Ｔｈｅｏｒｅｍ 1 )ａｎｄｔｈｅＤｒａｚｉｎｉｎｖｅｒｓｅｏｆａｍａｔｒｉｘｏｖｅｒｔｈｅｃｏｍｐ…     

含x^5项强非线性系统的共振解和亚谐解

本文应用强非线性系统频闪法,对公式进行适当变换,求出了方程x+ax+βx~3+γx~5=ε(-μx+δcosΩt)的共振解及亚谐解存在的条件及其解析表达式,并与数值计算结果作了比较.     

态射的加权Moore-Penrose逆

讨论带有对合＊的范畴C中具有满单分解态射的加权Moore-Penrose逆,给出了具有满单分解态射的加权Moore-Penrose逆存在的几个充要条件以及具有满单分解的加权Moore-Penrose逆的表达式,所得结论和公式包括了关于广义Moore-Penrose逆中的结果.     

Some issues surrounding the use of algebraic calculators in traditional examinations

Algebraic calculators, with symbolic manipulation capabilities, are tools in some classrooms for 14–19-year-olds and their use in learning and teaching mathematics is likely to increase over time. At the time of writing their use in traditional examinations is permitted in Austria, France and the USA. Several other countries have considered their possible use in examinations. In the UK several government initiated working groups considered the implications of their use in academic mathematics examinations for 18-year-old students. The reports of these working groups have not been published. An overview of the issues they addressed is presented here and implications for future examinations are considered.