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所在平台: Udemy |
课程主页: https://www.udemy.com/course/zsigmondys-theorem/
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课程名称：西格蒙特（Zsigmondy）定理 课程概述： 本课程将以证明数论中的西格蒙特定理为主线，并展示如何运用该定理解决一些在数学奥林匹克竞赛中具有挑战性的题目。为达成此目标，我们将首先深入理解高中代数中最核心但常被高中忽略的主题——分圆多项式。分圆多项式看似专业，实则是高中代数核心内容（如平方差、立方差因式分解）的基础。分圆多项式提供了 $x^n - 1$ 的一种因式分解。当 $n=2$ 时，它就是平方差公式。通过简单变换，令 $x$ 为 $x/y$，即可得到 $x^2 - y^2$。本课程对于认真学习高中数学或致力于奥林匹克竞赛的学生来说，将具有极高的价值。本课程的一大亮点是，讲师将与学生共同学习西格蒙特定理的证明过程，让学生体验自主学习，无需他人额外指导。 学习目标： * 理解分圆多项式的概念及其在代数中的重要性。 * 掌握分圆多项式在 $x^n - 1$ 因式分解中的应用。 * 学习并证明西格蒙特定理。 * 运用西格蒙特定理解决数学奥林匹克竞赛难题。 * 培养自主学习能力。
The story line that guides us is proving a theorem of Zsigmondy in number theory and seeing how it can be used to solve maths olympiad problems that would otherwise be quite difficult. To achieve this goal we first understand what I consider to be the most central topic in high school algebra which is omitted in high schools: cyclotomic polynomials. This sounds specialised but this is at the heart of all the algebra learned at high school such as factorising a difference of 2 squares or cubes. The cyclotomic polynomials gives a factorisation of x^n-1. When n is 2, this is just the difference of 2 squares. If you let the x be x/y then you really get x^2-y^2 after some easy manipulation. (x^n means x to the power of n)These lessons will be a very valuable part of a serious high school maths student or olympian.One of the really interesting features of this course is that the instructor learns the proof of the Zsigmondy Theorem with the students and you get to see how to educate yourself without further need to be taught.
