# Difference between revisions of "Polynomial ring"

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− | Given a (commutative) [[ring]] <math>R</math>, the polynomial ring <math>R[x]</math> is, informally, "the ring of all polynomials in <math>x</math> with coefficients in <math>R</math>." | + | Given a (commutative) [[ring]] <math>R</math>, the '''polynomial ring''' <math>R[x]</math> is, informally, "the ring of all polynomials in <math>x</math> with coefficients in <math>R</math>." |

+ | <cmath>R[x]=\left\lbrace\sum_{i=0}^\infty a_ix^i\mid a_i\in R\right\rbrace</cmath> | ||

+ | |||

+ | ==Formal Definition== | ||

+ | |||

+ | We can rigorously define <math>R[x]</math> to be the set of all sequences of elements of <math>R</math> with only finitely many terms nonzero: | ||

+ | <cmath>R[x] = \{(a_0,a_1,a_2,\ldots)|\text{the set }\{i|a_i\neq 0\} \text{ is finite }\}</cmath> | ||

+ | The we call the elements of <math>R[x]</math> '''polynomials''' (over <math>R</math>). For a polynomial <math>p=(a_0,a_1,a_2,\ldots)</math>, the terms <math>a_0,a_1,a_2,\ldots</math> are called the '''coefficients''' of <math>p</math>. | ||

+ | |||

+ | For example, <math>(0,0,0,\ldots), (0,1,0,0,\ldots), (1,4,0,3,0,0,\ldots)</math> would be considered polynomials, but <math>(1,1,1,1,\ldots)</math> 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 <math>(a_0,a_1,a_2,\ldots)</math> by <math>a_0+a_1x+a_2x^2+\cdots</math>. For instance we would write: | ||

+ | |||

+ | <math> | ||

+ | \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*} | ||

+ | </math> | ||

+ | |||

+ | Typically, we repress the terms with coefficient <math>0</math> and we do not write the coefficient on terms with coefficient <math>1</math>. 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: | ||

+ | |||

+ | <math> | ||

+ | \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*} | ||

+ | </math> | ||

+ | |||

+ | It is important to note at this point that '<math>x</math>' 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 <math>R</math>. Furthermore, a polynomial is '''not''' a function. | ||

+ | |||

+ | One can now define addition and multiplication in <math>R[x]</math> in the 'obvious' way: | ||

+ | *<cmath>\sum_i a_ix^i + \sum_i b_ix^x = \sum_i (a_i+b_i)x^i</cmath> | ||

+ | *<cmath>\left(\sum_i a_ix^i\right)\cdot \left(\sum_j b_jx^j\right) = \sum_k\left(\sum_{i=0}^k a_ib_{k-i}\right)x^k</cmath> | ||

+ | It is now a simple matter to verify that <math>R[x]</math> indeed forms a commutative ring under these operations. This ring has additive identity <math>0=(0,0,0,\ldots)</math> and multiplicative identity <math>1 = (1,0,0,\ldots)</math>. | ||

− | < | + | <math>R</math> can be thought of as a [[subring]] of <math>R[x]</math> via the embedding <math>r\mapsto (r,0,0,\ldots)</math>. |

{{stub}} | {{stub}} | ||

+ | [[Category:Ring theory]] |

## Revision as of 16:43, 26 March 2009

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 .

*This article is a stub. Help us out by expanding it.*