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# 数学代写|抽象代数代写Abstract Algebra代考|MATH355 Weird Dice: An Application of Unique Factorization

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## 数学代写|抽象代数代写Abstract Algebra代考|Weird Dice: An Application of Unique Factorization

I EXAMPLE 12 Consider an ordinary pair of dice whose faces are labeled 1 through 6 . The probability of rolling a sum of 2 is $1 / 36$, the probability of rolling a sum of 3 is 2/36, and so on. In a 1978 issue of Scientific American, Martin Gardner remarked that if one were to label the six faces of one cube with the integers $1,2,2,3,3,4$ and the six faces of another cube with the integers $1,3,4,5,6,8$, then the probability of obtaining any particular sum with these dice (called Sicherman dice) would be the same as the probability of rolling that sum with ordinary dice (i.e., $1 / 36$ for a 2,2/36 for a 3, and so on). See Figure 17.1. In this example, we show how the Sicherman labels can be derived, and that they are the only possible such labels besides 1 through 6 . To do so, we utilize the fact that $Z[x]$ has the unique factorization property.

To begin, let us ask ourselves how we may obtain a sum of 6, say, with an ordinary pair of dice. Well, there are five possibilities for the two faces: $(5,1),(4,2),(3,3),(2,4)$, and $(1,5)$. Next we consider the product of the two polynomials created by using the ordinary dice labels as exponents:
$$\left(x^6+x^5+x^4+x^3+x^2+x\right)\left(x^6+x^5+x^4+x^3+x^2+x\right) .$$
Observe that we pick up the term $x^6$ in this product in precisely the following ways: $x^5 \cdot x^1, x^4 \cdot x^2, x^3 \cdot x^3, x^2 \cdot x^4, x^1 \cdot x^5$. Notice the correspondence between pairs of labels whose sums are 6 and pairs of terms whose products are $x^6$. This correspondence is one-to-one, and it is valid for all sums and all dice-including the Sicherman dice and any other dice that yield the desired probabilities. So, let $a_1, a_2, a_3, a_4, a_5, a_6$ and $b_1, b_2, b_3, b_4, b_5, b_6$ be any two lists of positive integer labels for the faces of a pair of cubes with the property that the probability of rolling any particular sum with these dice (let us call them weird dice) is the same as the probability of rolling that sum with ordinary dice labeled 1 through 6 . Using our observation about products of polynomials, this means that
\begin{aligned} &\left(x^6+x^5+x^4+x^3+x^2+x\right)\left(x^6+x^5+x^4+x^3+x^2+x\right) \ &=\left(x^{a_1}+x^{a_2}+x^{a_3}+x^{a_4}+x^{a_5}+x^{a_6}\right) \ &\left(x^{b_1}+x^{b_2}+x^{b_3}+x^{b_4}+x^{b_5}+x^{b_6}\right) \end{aligned}

## 数学代写|抽象代数代写Abstract Algebra代考|Irreducibles, Primes

An integer $\mathrm{n}$ with decimal representation $a_k a_{k-1} \cdots a_0$ is divisible by 9 if and only if $a_k+a_{k-1}+\cdots+a_0$ is divisible by 9 . To verify this, observe that $n=a_k 10^k+a_{k-1} 10^{k-1}+\cdots+a_0$. Then, letting $\alpha$. denote the natural homomorphism from $\mathrm{Z}$ to $Z_9$ [in particular, $\alpha(10)=1$ ], we note that $\mathrm{n}$ is divisible by 9 if and only if
\begin{aligned} 0=\alpha(n) &=\alpha\left(a_k\right)(\alpha(10))^k+\alpha\left(a_{k-1}\right)(\alpha(10))^{k-1}+\cdots+\alpha\left(a_0\right) \ &=\alpha\left(a_k\right)+\alpha\left(a_{k-1}\right)+\cdots+\alpha\left(a_0\right) \ &=\alpha\left(a_k+a_{k-1}+\cdots+a_0\right) \end{aligned}
But $\alpha\left(a_k+a_{k-1}+\cdots+a_0\right)=0$ is equivalent to $a_k+a_{k-1}+\cdots+a_0$ being divisible by 9 .
The next example illustrates the value of the natural homomorphism given in Example $1 .$

In the preceding two chapters, we focused on factoring polynomials over the integers or a field. Several of those results-unique factorization in $Z[x]$ and the division algorithm for $F[x]$, for instance – are natural counterparts to theorems about the integers. In this chapter and the next, we examine factoring in a more abstract setting.
Definition Associates, Irreducibles, Primes Elements $a$ and $b$ of an integral domain $D$ are called associates if $a=u b$, where $u$ is a unit of $D$. A nonzero element a of an integral domain $D$ is called an irreducible if $a$ is not a unit and, whenever $b, c \in D$ with $a=b c$, then $b$ or $c$ is a unit. A nonzero element $a$ of an integral domain $D$ is called a prime if $a$ is not a unit and $a \mid b c$ cimplies $a \mid b$ or $a \mid c$.

Roughly speaking, an irreducible is an element that can be factored only in a trivial way. Notice that an element a is a prime if and only if $\langle a\rangle$ is a prime ideal.
Relating the definitions above to the integers may seem a bit confusing, since in Chapter 0 we defined a positive integer to be a prime if it satisfies our definition of an irreducible, and we proved that a prime integer satisfies the definition of a prime in an integral domain (Euclid’s Lemma). The source of the confusion is that in the case of the integers, the concepts of irreducibles and primes are equivalent, but in general, as we will soon see, they are not.
The distinction between primes and irreducibles is best illustrated by integral domains of the form $Z[\sqrt{d}>]={a+b \sqrt{d} \mid a, b \in Z}$ , where $d$ is not 1 and is not divisible by the square of a prime.
(These rings are of fundamental importance in number theory.) To analyze these rings, we need a convenient method of determining their units, irreducibles, and primes. To do this, we define a function $N$, called the norm, from $Z[\sqrt{d}>]$ into the nonnegative integers by $N(a+b \sqrt{d})=\left|a^2-d b^2\right| .$ We leave it to the reader (Exercise 1) to verify the following four properties: $N(x)=0$ if and only if $x=0 ; N(x y)=N(x) N(y)$ for all $x$ and $y ; x$ is a unit if and only if $N(x)=1 ;$ and, if $N(x)$ is prime, then $x$ is irreducible in $Z[\sqrt{d}>] .$

## 数学代写|抽象代数代写抽象代数代考|怪异的骰子:唯一因子分解的应用

.

$$\left(x^6+x^5+x^4+x^3+x^2+x\right)\left(x^6+x^5+x^4+x^3+x^2+x\right) .$$

\begin{aligned} &\left(x^6+x^5+x^4+x^3+x^2+x\right)\left(x^6+x^5+x^4+x^3+x^2+x\right) \ &=\left(x^{a_1}+x^{a_2}+x^{a_3}+x^{a_4}+x^{a_5}+x^{a_6}\right) \ &\left(x^{b_1}+x^{b_2}+x^{b_3}+x^{b_4}+x^{b_5}+x^{b_6}\right) \end{aligned}

## 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。