Difference between revisions of "Polynomial ring"

(more formal definition? do we need to prove ringness? also {{stub}})
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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>."  
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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>."  
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<cmath>R[x]=\left\lbrace\sum_{i=0}^\infty a_ix^i\mid a_i\in R\right\rbrace</cmath>
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==Formal Definition==
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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:
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<cmath>R[x] = \{(a_0,a_1,a_2,\ldots)|\text{the set }\{i|a_i\neq 0\} \text{ is finite }\}</cmath>
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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>.
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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.
  
<cmath>R[x]=\left\lbrace\sum_{i=0}^\infty a_ix^i\mid a_i\in R\right\rbrace</cmath>
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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:
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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:
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<math>
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\begin{align*}
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(0,0,0,\ldots) &= 0+0x+0x^2+\cdots\\
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(0,1,0,0,\ldots) &= 0+1x+0x^2+0x^3+\cdots\\
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(1,4,0,3,0,0,\ldots) &= 1+4x+0x^2+3x^3+0x^4+0x^5+\cdots
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\end{align*}
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</math>
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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:
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<math>
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\begin{align*}
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(0,0,0,\ldots) &= 0\\
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(0,1,0,0,\ldots) &= x\\
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(1,4,0,3,0,0,\ldots) &= 3x^3+4x+1
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\end{align*}
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</math>
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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.
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One can now define addition and multiplication in <math>R[x]</math> in the 'obvious' way:
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*<cmath>\sum_i a_ix^i + \sum_i b_ix^x = \sum_i (a_i+b_i)x^i</cmath>
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*<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>
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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>.
  
<!-- do we need to prove the ringness of R[x]?-->
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<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}}
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[[Category:Ring theory]]

Revision as of 16:43, 26 March 2009

Given a (commutative) ring $R$, the polynomial ring $R[x]$ is, informally, "the ring of all polynomials in $x$ with coefficients in $R$." \[R[x]=\left\lbrace\sum_{i=0}^\infty a_ix^i\mid a_i\in R\right\rbrace\]

Formal Definition

We can rigorously define $R[x]$ to be the set of all sequences of elements of $R$ with only finitely many terms nonzero: \[R[x] = \{(a_0,a_1,a_2,\ldots)|\text{the set }\{i|a_i\neq 0\} \text{ is finite }\}\] The we call the elements of $R[x]$ polynomials (over $R$). For a polynomial $p=(a_0,a_1,a_2,\ldots)$, the terms $a_0,a_1,a_2,\ldots$ are called the coefficients of $p$.

For example, $(0,0,0,\ldots), (0,1,0,0,\ldots), (1,4,0,3,0,0,\ldots)$ would be considered polynomials, but $(1,1,1,1,\ldots)$ 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 $(a_0,a_1,a_2,\ldots)$ by $a_0+a_1x+a_2x^2+\cdots$. 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 $0$ and we do not write the coefficient on terms with coefficient $1$. 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 '$x$' 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 $R$. Furthermore, a polynomial is not a function.

One can now define addition and multiplication in $R[x]$ in the 'obvious' way:

  • \[\sum_i a_ix^i + \sum_i b_ix^x = \sum_i (a_i+b_i)x^i\]
  • \[\left(\sum_i a_ix^i\right)\cdot \left(\sum_j b_jx^j\right) = \sum_k\left(\sum_{i=0}^k a_ib_{k-i}\right)x^k\]

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

$R$ can be thought of as a subring of $R[x]$ via the embedding $r\mapsto (r,0,0,\ldots)$.

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