# Polynomial ring

Given a (commutative) ring , the **polynomial ring** is, informally, "the ring of all polynomials in with coefficients in ."

## Formal Definition

We can rigorously define to be the set of all sequences of elements of with only finitely many terms nonzero:
The we call the elements of **polynomials** (over ). For a polynomial , the terms are called the **coefficients** of .

For example, would be considered polynomials, but would not be.

At this point, our formal definition of a polynomial may seem unrelated to our intuitive notion of a polynomial. To relate these two concepts, we introduce the following notation:

We will denote the polynomial by . For instance we would write:

$\begin{align*} (0,0,0,\ldots) &= 0+0x+0x^2+\cdots\\ (0,1,0,0,\ldots) &= 0+1x+0x^2+0x^3+\cdots\\ (1,4,0,3,0,0,\ldots) &= 1+4x+0x^2+3x^3+0x^4+0x^5+\cdots \end{align*}$ (Error compiling LaTeX. ! Package amsmath Error: \begin{align*} allowed only in paragraph mode.)

Typically, we repress the terms with coefficient and we do not write the coefficient on terms with coefficient . We also do not care about the order in which the terms are written, and indeed often list them in descending order of power. So we would write:

$\begin{align*} (0,0,0,\ldots) &= 0\\ (0,1,0,0,\ldots) &= x\\ (1,4,0,3,0,0,\ldots) &= 3x^3+4x+1 \end{align*}$ (Error compiling LaTeX. ! Package amsmath Error: \begin{align*} allowed only in paragraph mode.)

It is important to note at this point that '' is only a symbol, it has no independent meaning, and in particular it is **not** a variable, i.e. is does **not** represent an element of . Furthermore, a polynomial is **not** a function.

One can now define addition and multiplication in in the 'obvious' way:

It is now a simple matter to verify that indeed forms a commutative ring under these operations. This ring has additive identity and multiplicative identity .

can be thought of as a subring of via the embedding .

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