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数学代写|现代代数代考Modern Algebra代写|Math4120 The fundamental theorem of arithmetic

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数学代写|现代代数代考Modern Algebra代写|The fundamental theorem of arithmetic

We proved above that every natural number could be factored as a product of primes. But we want more than existence, we want uniqueness. We need to prove that there is only one way that it can be factored as a product of primes.

The unique factorization theorem, a.k.a., the fundamental theorem of arithmetic. Now, in order to make this general statement valid we have to extend a little bit what we mean by a product. For example, how do you write a prime number like 7 as a product of primes? It has to be written as the product 7 of only one prime. So we will have to accept a single number as being a product of one factor.

Even worse, what about 1 ? There are no primes that divide 1 . One solution is to accept a product of no factors as being equal to 1. It’s actually a reasonable solution to define the empty product to be 1 , but until we find another need for an empty product, let’s wait on that and restrict this unique factorization theorem to numbers greater than 1 . So, here’s the statement of the theorem we want to prove.

Theorem $1.46$ (Unique factorization theorem). Each integer $n$ greater than 1 can be uniquely factored as a product of primes. That is, if $n$ equals the product $p_1 p_2 \cdots p_r$ of $r$ primes, and it also equals the product $q_1 q_2 \cdots q_s$ of $s$ primes, then the number of factors in the two products is the same, that is $r=s$, and the two lists of primes $p_1, p_2, \ldots, p_r$ and $q_1, q_2, \ldots, q_s$ are the same apart from the order the listings.

We’ll prove this by using the strong form of mathematical induction. The form that we’ll use is this:
In order to prove a statement $S(n)$ is true for all numbers, prove that $S(n)$ follows from the assumption that $S(k)$ is true for all $k<n$.

数学代写|现代代数代考Modern Algebra代写|Polynomials

We’ll frequently use polynomials in our study of fields and rings. We’ll only consider polynomials with coefficients in fields and commutative rings, not with coefficients in noncommutative rings.

We won’t formally define polynomials. For now, we’ll only look at polynomials in one variable $x$, but later in section $3.10 .4$ we’ll look at polynomials in two or more variables.

Informally a polynomial $f(x)$ with coefficients in a commutative ring $R$ is an expression
$$f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_1 x+a_0$$
where each coefficient $a_i \in R$. We’ll assume that the leading coefficient $a_n$ is not zero so that $\operatorname{deg} f$, the degree of the polynomial, is $n$. When $a_n$ is zero, the polynomial is called a monic polynomial.

It’s convenient to denote a polynomial either by $f$ or by $f(x)$. If the variable $x$ is referred to somewhere nearby, then I’ll use $f(x)$, otherwise I’ll just use $f$. For instance, if I want to multiply two polynomials $f$ and $g$ together, I’ll write $f g$, but if I want two multiply $f$ by $x^2-3 x+2$, I’ll write $f(x)\left(x^2-3 x+2\right)$ or $f(x) \cdot\left(x^2-3 x+2\right)$.

A root of a polynomial is an element $a$ of $R$ such that $f(a)=0$, that is, it’s a solution of the polynomial equation $f(x)=0$.

The set of all polynomials with coefficients in a commutative ring $R$ is denoted $R[x]$. It has addition, subtraction, and multiplication, and satisfies the requirements of a ring, that is, it has addition, subtraction, and multiplication with the usual properties. $R[x]$ is called the ring of polynomials with coefficients in $R$. Note that $R[x]$ doesn’t have reciprocals even when $R$ is a field, since $x$ has no inverse in $R[x]$. Therefore, $R[x]$ is not a field. Nonetheless, the ring $R$ is a subring of the ring $R[x]$ since we can identify the constant polynomials as the elements of $R$.

现代代数代写

数学代写|现代代数代考Modern Algebra代写|Polynomials

$$f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_1 x+a_0$$

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MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。