# Abstract Nonsense

## Polynomial Rings (Pt. I)

Point of Post: In this post we define the ring $R[x]$ of polynomials in the determinate $x$ over a ring $R$ as well as then describing the rings $R[x_1,\cdots,x_n]$ for indeterminates $x_1,\cdots,x_n$. We then discuss some of the properties (although not a lot, since this particular ring will hold a lot of our interests in the future) of this ring.

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Motivation

We have discussed a few ways of taking given rings $R$ and constructing new rings out of them, such as forming the matrix ring $\text{Mat}_n(R)$ and the $n$-fold product ring $R^n$. In this post we discuss yet another way of forming a ring in a ‘functorial’, namely the ring $R[x]$ of polynomials whose coefficients lie in $R$. It’s funny that this is perhaps the ring construction that needs the least motivation for beginners, but for all the wrong reasons. Some people would say “Oh, polynomials! We’ve been using those since middle school, of course they’re important!” but the real reason why polynomial rings are important transcends their ubiquity. Indeed, it could be said that algebra began with the study of polynomial rings. One of the first considerations of an undergraduate algebra student, the first thing thrown at them as ‘wow inducing’ is the inability for form a ‘quintic’ equation (i.e. a diluted form of the Abel-Ruffini theorem). While slightly overplayed this is a very important result and concepts since it’s question (does there exist a quintic equation) and the subsequent journey it motivated could be thought of as the first attempts at a modern theory of algebra. Since there has been a veritable blossoming of algebraic and algebraic related fields, and centerpiece to most of them are rings of polynomials. Indeed, one could say that modern number theory (in the most basic of its definitions) could be described not only as the study of the ring $R$ but of the ring $\mathbb{Z}[x]$, the fields of algebraic geometry and commutative algebra have rings of the form $k[x_1,\cdots,x_n]$ where $k$ is some field as their main player, they are very important in the modern study of functional analysis where the polynomial ring $k[T]$ for some linear operator $T$ is of key importance, etc.

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Perhaps one of the key aspects to this importance is the somewhat dual nature a polynomial ring has. In particular, strictly speaking, the elements of $R[x]$ are ‘formal symbols’ where the ‘$x$‘s are given more the role of convenient placeholder, in other words in this sense one could equally well think of elements of $R[x]$ as tuples in $R^{\mathbb{N}}$ with finite support which we multiply in a ‘funny way’. That said, every polynomial $p(x)=a_0+\cdots+a_nx^n\in R[x]$ naturally induces a function $R\to R$ given by $p(r)=a_0+\cdots+a_n r^n$. Given zero thought and with years of inadvertent inculcation one may go “What do you mean dual concepts, they are the same thing!”, but of course, this is totally wrong. For example, as formal symbols the elements $1$ and $1+x+x^2$ are different elements of $\mathbb{Z}_2[x]$, yet as one can check the induced function $\mathbb{Z}_2\to\mathbb{Z}_2:x\mapsto 1+x+x^2$ is the same as the function $\mathbb{Z}_2\to\mathbb{Z}_2:x\mapsto 1$. That said, they are dual in the sense that information about $R[x]$ when thought about as a set of functions gives information when thought about as a ring of formal symbols and vice versa. For example, we will be able to tell a lot about the ring structure of $R[x]$ by considering the evaluation homomorphism $\text{ev}_r:R[x]\to R:p\mapsto p(r)$. Conversely, we will prove that under certain conditions the functions $p(x)$ has a root $\alpha\in R$ if and only $p(x)=(x-\alpha)q(x)$ for some $q(x)\in R[x]$.

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One should note that in the below we restrict out attention to when $R$ is commutative and unital, only because it makes things much more complicated to assume otherwise, and in all reasonable circumstances when we shall want to consider $R[x]$ we shall readily see that $R$ is commutative and unital.

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Polynomial Rings

Let $R$ be a commutative unital ring. We define the polynomial ring in the indeterminate $x$ over $R$, denoted $R[x]$, to be the set of all formal sums of the form $\displaystyle \sum_{n\in\mathbb{N}}r_nx^n+r_0$ where $r_n\in R$ and $r_n=0$ for all but finitely many $n\in\mathbb{N}$. To make things easier we define $x^0=1$ and omit terms if the coefficient is zero. So, for example $3+0x+0x^2+5x^3+0x^4+5x^5+0x^6+0x^7+\cdots$ is alternatively denoted $3+5x^3+5x^5$ or $3x^0+5x^3+5x^5$ in $\mathbb{Z}[x]$. As a convention if we write a polynomial in the ‘finite sum’ form $a_0+\cdots+a_nx^n$ it is assumed that $a_n\ne 0$ if $n>0$. We  define that two polynomials $\displaystyle \sum_{n\in\mathbb{N}}r_nx^n+r_0$ and $\displaystyle \sum_{n\in\mathbb{N}}s_nx^n+s_0$ are equal if $a_k=s_k$ for every $k\in\mathbb{N}\cup\{0\}$. Or, said differently we define two polynomials $a_0+\cdots+a_nx^n$ and $b_0+\cdots+b_mx^m$ are equal if $m=n$ and $a_k=b_k$ for $k=0,\cdots,n$. For a polynomial $a_0+\cdots+a_nx^n$ we call $a_n$ the leading coefficient and $n$ the degree and denote it $\deg(a_0+\cdots+a_nx^n)$. We call a polynomial monic if the leading coefficient is $1$.

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We define addition in $R[x]$ by defining

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$\displaystyle \left(\sum_{n\in\mathbb{N}}r_nx_n+r_0\right)+\left(\sum_{n\in\mathbb{N}}s_nx^n+s_0\right)=\sum_{n\in\mathbb{N}}(r_n+s_n)x^n+(r_0+s_0)$

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So that if $n=m$ then

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$(a_0+\cdots+a_nx^n)+(b_0+\cdots+b_mx^m)=(a_0+b_0)+\cdots+(a_n+b_n)x^n$

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and if $n>m$ then

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$(a_0+b_0)+\cdots+(a_m+b_m)x^m+a_{m+1}x^{m+1}+\cdots+a_nx^n$

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We define the product of two polynomials by

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$\displaystyle \left(\sum_{n\in\mathbb{N}}r_nx_n+r_0\right)\left(\sum_{n\in\mathbb{N}}s_nx^n+s_0\right)=\sum_{n\in\mathbb{N}}\sum_{j=0}^{n}r_js_{n-j}x^n+r_0s_0$

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Viewing this differently, once we define $(ax^n)(bx^m)$ we will have already decided the definition of the product on all of $R[x]$ since the $x^n$s form a ‘basis’ and we want multiplication to be distributive. So, our definition of product can be seen by defining $(ax^n)(bx^m)=abx^{n+m}$ and then extending this to all of $R[x]$ by distributivity–colloquially we ‘expand and collect terms’. It is a laborious yet straightforward task to verify that with these definitions that $R[x]$ is a commutative unital ring with identity equal to $1=1x^0$.

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We see by definition that there is a natural embedding $R\hookrightarrow R[x]$ defined by $r\mapsto r=rx^0$. We call the image of this map the constant polynomials and as observed see that they form a subring canonically isomorphic to $R$. Consequently, we shall identify $R$ with the constant polynomials, the confusion between this identification shall be very minimal (opposed to the identifications in other areas of mathematics).

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References:

1. Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.

2. Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul. Basic Abstract Algebra. Cambridge [Cambridgeshire: Cambridge UP, 1986. Print.

July 19, 2011 -

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