cyclotomic

简明释义

英[ˌsaɪkləˈtɒmɪk]美[ˌsaɪkləˈtɒmɪk]

adj. 分圆的

英英释义

单词用法

同义词

反义词

例句

1.The cyclotomic continued proportion is impossible to furnish Qing-times series with a solid basis.

割圆连比例不可能为清代级数论提供坚实的基础。

2.An algorithm for cyclotomic polynomial test is presented based on that all the roots of a cyclotomic polynomial are unity. The algorithm is directed. simple and realized easily.

本文利用分圆多项式的所有根是单位根的性质，直接从给定的多项式入手，提出了判定一个多项式是否为分圆多项式的算法．算法简单明了，易于实现．。

3.At the same time, Archimedes gives an exact proof on the formula of area of circle, with the method of exhaustion method depending on the Cyclotomic.

刘徽之前，希腊的阿基米德用穷竭法也证明了圆的面积公式。

4.Cyclotomic continued proportion was the principal method for studies on infinite series in the Qing Dynasty, which we still have to go into a step further.

割圆连比例曾是清代无穷级数研究中所用的主要方法，但对它的研究目前还不够充分。

5.An algorithm for cyclotomic polynomial test is presented based on that all the roots of a cyclotomic polynomial are unity. The algorithm is directed. simple and realized easily.

本文利用分圆多项式的所有根是单位根的性质，直接从给定的多项式入手，提出了判定一个多项式是否为分圆多项式的算法．算法简单明了，易于实现．。

6.In this paper, I first give two methods with which we can determine all the power integral bases in maximal real subfields of cyclotomic fields.

本文首先给出了确定分圆域的极大实子域的幂元整基的两种方法；

7.Mathematicians often explore the properties of cyclotomic extensions.

数学家们常常探索循环扩展的性质。

8.The roots of cyclotomic equations are closely related to the concept of primitive roots.

循环方程的根与原根的概念密切相关。

9.Understanding cyclotomic numbers is essential for advanced algebra courses.

理解循环数对于高级代数课程至关重要。

10.In algebra, cyclotomic polynomials play a crucial role in factorization.

在代数中，循环多项式在因式分解中起着关键作用。

11.The study of cyclotomic fields has significant implications in number theory.

对循环域的研究在数论中具有重要意义。

作文

In the field of mathematics, particularly in number theory and algebra, the term cyclotomic refers to a special class of polynomials known as cyclotomic polynomials. These polynomials are defined as the minimal polynomials of the primitive roots of unity. A primitive root of unity is a complex number that, when raised to a certain integer power, equals one. The significance of cyclotomic polynomials lies in their ability to factorize integers and provide insights into the structure of various algebraic equations. For example, the n-th cyclotomic polynomial, denoted by φ(n), can be expressed as the product of linear factors over the complex numbers, which reveals the roots of unity associated with that polynomial. The study of cyclotomic fields, which are extensions of the rational numbers obtained by adjoining a primitive root of unity, has profound implications in both theoretical and applied mathematics. These fields help mathematicians understand Galois groups, which are fundamental to solving polynomial equations. The Galois group of a cyclotomic field can often be described in terms of the symmetries of the roots of unity, leading to deeper insights into the nature of solutions to polynomial equations. Moreover, cyclotomic numbers and their properties have applications in various areas, including cryptography, coding theory, and even signal processing. For instance, the discrete Fourier transform, which is crucial in digital signal processing, can be understood through the lens of cyclotomic fields and polynomials. The efficient computation of the discrete Fourier transform relies on the properties of cyclotomic numbers, demonstrating their practical importance in modern technology. In addition to their mathematical significance, cyclotomic concepts also extend into the realm of abstract algebra. The structure of the group of units modulo n, which consists of the integers coprime to n, can be analyzed using cyclotomic fields. This connection illustrates how number theory and group theory intersect, providing a rich tapestry of relationships among different branches of mathematics. Furthermore, the study of cyclotomic extensions leads to important results in algebraic number theory. For instance, the properties of cyclotomic fields allow mathematicians to explore the distribution of prime numbers and investigate the nature of Diophantine equations. The connections between cyclotomic fields and class numbers highlight the intricate relationships between algebraic structures and number-theoretic properties. In conclusion, the term cyclotomic encapsulates a wealth of mathematical concepts that bridge various disciplines within mathematics. From its origins in the study of roots of unity to its applications in modern technology and theoretical exploration, understanding cyclotomic polynomials and fields is essential for anyone delving into advanced mathematics. As we continue to explore the depths of this fascinating area, the implications of cyclotomic theory will undoubtedly lead to new discoveries and innovations in both mathematics and its applications in the real world.

在数学领域，特别是数论和代数中，术语cyclotomic指的是一类特殊的多项式，称为循环多项式。这些多项式被定义为原根单位的最小多项式。原根单位是一个复数，当其被提升到某个整数次方时等于一。cyclotomic多项式的重要性在于它们能够对整数进行因式分解，并提供对各种代数方程结构的洞察。例如，第n个cyclotomic多项式，记作φ(n)，可以表示为线性因子的乘积，揭示与该多项式相关的单位根。对cyclotomic域的研究，即通过连接原根单位来获得有理数的扩展，在理论和应用数学中都有深远的影响。这些域帮助数学家理解伽罗瓦群，这是解决多项式方程的基础。cyclotomic域的伽罗瓦群通常可以用单位根的对称性来描述，从而深入了解多项式方程解的性质。此外，cyclotomic数字及其属性在多个领域都有应用，包括密码学、编码理论甚至信号处理。例如，离散傅里叶变换在数字信号处理中至关重要，可以通过cyclotomic域和多项式的视角来理解。离散傅里叶变换的高效计算依赖于cyclotomic数字的性质，展示了它们在现代技术中的实际重要性。除了数学意义外，cyclotomic概念还延伸到抽象代数的领域。模n的单位群的结构，由与n互质的整数组成，可以利用cyclotomic域进行分析。这种联系说明了数论和群论之间的交汇，为不同数学分支之间提供了丰富的关系。此外，cyclotomic扩展的研究在代数数论中导致了重要的结果。例如，cyclotomic域的性质使数学家能够探索素数的分布并研究丢番图方程的性质。cyclotomic域和类数之间的联系突显了代数结构和数论性质之间的复杂关系。总之，术语cyclotomic概括了丰富的数学概念，跨越数学的各个学科。从其在单位根研究中的起源到其在现代技术和理论探索中的应用，理解cyclotomic多项式和域对于深入研究高级数学的人来说至关重要。随着我们继续探索这一迷人的领域，cyclotomic理论的影响无疑将引领我们在数学及其在现实世界中的应用方面发现新的发现和创新。