Difference between revisions of "Polynomial ring"

(Formal definition)
(added stuff; corrected some stuff)
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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 [[ring]] <math>R</math>, the '''polynomial ring''' <math>R[x]</math> is, informally, "the ring of all polynomials in a [[commutative]] <math>x</math> with coefficients in <math>R</math>." That is, it is the ring of all sums of the form
<cmath>R[x]=\left\lbrace\sum_{i=0}^\infty a_ix^i\mid a_i\in R\right\rbrace</cmath>
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<cmath> \sum_{k=0}^N a_k x^k </cmath>
 +
where <math>N</math> is a [[nonnegative integer]] that varies from sum to sum.
 +
 
 +
The ring <math>R[x]</math> is also an <math>R</math>-module.
  
 
==Formal Definition==
 
==Formal Definition==
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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>.
 
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.
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For example, <math>(0,0,0,\ldots), (0,1,0,0,\ldots), (1,4,0,3,0,0,\ldots)</math> are polynomials, but <math>(1,1,1,1,\ldots)</math> is not a polynomial.
 
 
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:
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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 some notation.
  
<math>
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We 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 write:
 +
<cmath>
 
\begin{align*}
 
\begin{align*}
 
(0,0,0,\ldots) &= 0+0x+0x^2+\cdots\\
 
(0,0,0,\ldots) &= 0+0x+0x^2+\cdots\\
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(1,4,0,3,0,0,\ldots) &= 1+4x+0x^2+3x^3+0x^4+0x^5+\cdots
 
(1,4,0,3,0,0,\ldots) &= 1+4x+0x^2+3x^3+0x^4+0x^5+\cdots
 
\end{align*}
 
\end{align*}
</math>
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</cmath>
 
 
 
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:
 
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:
 
+
<cmath>
<math>
 
 
\begin{align*}
 
\begin{align*}
 
(0,0,0,\ldots) &= 0\\
 
(0,0,0,\ldots) &= 0\\
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(1,4,0,3,0,0,\ldots) &= 3x^3+4x+1
 
(1,4,0,3,0,0,\ldots) &= 3x^3+4x+1
 
\end{align*}
 
\end{align*}
</math>
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</cmath>
 +
 
 +
We can now define addition and multiplication in <math>R[x]</math> in the canonical way:
 +
<cmath>
 +
\sum_i a_ix^i + \sum_i b_ix^x = \sum_i (a_i+b_i)x^i \\
 +
\biggl(\sum_i a_ix^i\biggr)\cdot \biggl(\sum_j b_jx^j\biggr) = \sum_k\biggl(\sum_{i=0}^k a_ib_{k-i}\biggr)x^k
 +
</cmath>
 +
It is now a simple matter to verify that <math>R[x]</math> indeed constitutes a ring under these operations, and that it is commutative when <math>R</math> is commutative. This ring has additive identity <math>0=(0,0,0,\ldots)</math> and multiplicative identity <math>1 = (1,0,0,\ldots)</math>.
 +
 
 +
The ring <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>.
 +
 
 +
For a polynomial <math>p = (a_0, \dotsc)</math>, the greatest integer <math>N</math> such that <math>a_N \neq 0</math> is called the ''degree of <math>p</math>''.  It is often denoted <math>\deg(p)</math>.
 +
 
 +
== Polynomials and Functions ==
 +
 
 +
Polynomials are not functions.  The symbol <math>x</math> does not represent a variable, but rather a commutative indeterminate, that is, a formal symbol that commutes with the elements of <math>R</math> and whose powers are independent of each other over <math>R</math>.  However, polynomials are associated with functions, called polynomial functions.  This is a historically important association: originally, the two concepts were almost inseperable.  Indeed, polynomial functions were almost certainly the first functions studied.  The concept of "function" was not articulated until the 12th to 14th centuries.  By Euler's time, "functions" were explicit rules of association built from elementary expressions, though Euler himself generalized the concept to what we now call continuous functions.  This began a long debate over how "function" should be defined that did not resolve until the 20th century, when the modern, abstract definition of "function" became standard.  The history of the concept of polynomial is more obscure, but they were almost certainly not divorced from their function roots until the beginnings of modern algebra in the 19th century.
 +
 
 +
Specifically, each element in <math>p \in R[x]</math> is ''associated'' with a function mapping <math>R</math> into itself; this function is evaluated at a value <math>a \in R</math> by replacing the symbol <math>x</math> with the element <math>a</math>.
 +
 
 +
More, formally, we can prove by induction on the degree of the elements of <math>R</math> that for any <math>a\in R</math> and any <math>p \in R[x]</math>, there is a unique element of <math>R</math> that is equivalent to <math>p</math> modulo <math>(x-a)</math>.  This unique element is sometimes denoted <math>p(a)</math>.  Thus we may associate each element <math>p \in R</math> with the mapping <math>a \mapsto p(a)</math> of <math>R</math> into itself.  (Alternatively, we can associate with each element <math>a\in R</math> a [[homomorphism]] of <math>R[x]</math> into <math>R</math> that is the composition of the canonical homomorphism of <math>R[x]</math> into <math>R[x]/(x-a)</math> and the canonical homomorphism of <math>R[x]/(x-a)</math> into <math>R</math>.)
 +
 
 +
It is important to note that although each polynomial in <math>p\in R</math> is associated with a function mapping <math>R</math> into itself, this function is not necessarily unique to <math>p</math>.  In particular, if <math>R</math> is [[finite]], then the set of functions mapping <math>R</math> into itself is finite, whereas <math>R[x]</math> is [[infinite]], so some functions must be associated with infinitely many different polynomials.  (In fact, it follows from the theory of [[coset]]s, applied to the additive groups involved, that ''every'' function that is associated with a polynomial must e associated with infinitely many polynomials.)
 +
 
 +
For example, if <math>R</math> is the ring of [[integer]]s modulo <math>p</math>, for <math>p</math> a [[prime number | prime]], then [[Fermat's Little Theorem]] states that the polynomials <math>x^p</math> and <math>x</math> are associated with the same functions mapping <math>R</math> into itself.
 +
 
 +
Nevertheless, in many infinite rings (such as the ring of integers), this association of polynomials with functions ''is'' unique.  In such contexts, the polynomials are often identified with their functions, by abuse of language.  The association of polynomials with functions is an important one: polynomials were first studied as polynomial functions, and indeed it was not until recently that functions gained their modern definition, quite divorced from polynomials.
 +
 
 +
== Finitude of Degree ==
  
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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"Polynomials of infinite degree" are properly called [[formal power series]].  The set of formal power series over a ring <math>R</math> constitutes a ring, denoted <math>R[[x]]</math>, of which the ring of polynomials is a subring.  In general, formal power series are not associated with mappings of <math>R</math> into itself, as infinitely iterated addition is not generally well-defined.
  
One can now define addition and multiplication in <math>R[x]</math> in the 'obvious' way:
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== See also ==
*<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>.
+
* [[Polynomial]]
 +
* [[Formal power series]]
  
{{stub}}
 
 
[[Category:Ring theory]]
 
[[Category:Ring theory]]

Revision as of 21:14, 27 March 2009

Given a ring $R$, the polynomial ring $R[x]$ is, informally, "the ring of all polynomials in a commutative $x$ with coefficients in $R$." That is, it is the ring of all sums of the form \[\sum_{k=0}^N a_k x^k\] where $N$ is a nonnegative integer that varies from sum to sum.

The ring $R[x]$ is also an $R$-module.

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, $(0,0,0,\ldots), (0,1,0,0,\ldots), (1,4,0,3,0,0,\ldots)$ are polynomials, but $(1,1,1,1,\ldots)$ is not a polynomial.

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 some notation.

We denote the polynomial $(a_0,a_1,a_2,\ldots)$ by $a_0+a_1x+a_2x^2+\cdots$. For instance, we 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*} 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*}

We can now define addition and multiplication in $R[x]$ in the canonical way: \[\sum_i a_ix^i + \sum_i b_ix^x = \sum_i (a_i+b_i)x^i \\ \biggl(\sum_i a_ix^i\biggr)\cdot \biggl(\sum_j b_jx^j\biggr) = \sum_k\biggl(\sum_{i=0}^k a_ib_{k-i}\biggr)x^k\] It is now a simple matter to verify that $R[x]$ indeed constitutes a ring under these operations, and that it is commutative when $R$ is commutative. This ring has additive identity $0=(0,0,0,\ldots)$ and multiplicative identity $1 = (1,0,0,\ldots)$.

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

For a polynomial $p = (a_0, \dotsc)$, the greatest integer $N$ such that $a_N \neq 0$ is called the degree of $p$. It is often denoted $\deg(p)$.

Polynomials and Functions

Polynomials are not functions. The symbol $x$ does not represent a variable, but rather a commutative indeterminate, that is, a formal symbol that commutes with the elements of $R$ and whose powers are independent of each other over $R$. However, polynomials are associated with functions, called polynomial functions. This is a historically important association: originally, the two concepts were almost inseperable. Indeed, polynomial functions were almost certainly the first functions studied. The concept of "function" was not articulated until the 12th to 14th centuries. By Euler's time, "functions" were explicit rules of association built from elementary expressions, though Euler himself generalized the concept to what we now call continuous functions. This began a long debate over how "function" should be defined that did not resolve until the 20th century, when the modern, abstract definition of "function" became standard. The history of the concept of polynomial is more obscure, but they were almost certainly not divorced from their function roots until the beginnings of modern algebra in the 19th century.

Specifically, each element in $p \in R[x]$ is associated with a function mapping $R$ into itself; this function is evaluated at a value $a \in R$ by replacing the symbol $x$ with the element $a$.

More, formally, we can prove by induction on the degree of the elements of $R$ that for any $a\in R$ and any $p \in R[x]$, there is a unique element of $R$ that is equivalent to $p$ modulo $(x-a)$. This unique element is sometimes denoted $p(a)$. Thus we may associate each element $p \in R$ with the mapping $a \mapsto p(a)$ of $R$ into itself. (Alternatively, we can associate with each element $a\in R$ a homomorphism of $R[x]$ into $R$ that is the composition of the canonical homomorphism of $R[x]$ into $R[x]/(x-a)$ and the canonical homomorphism of $R[x]/(x-a)$ into $R$.)

It is important to note that although each polynomial in $p\in R$ is associated with a function mapping $R$ into itself, this function is not necessarily unique to $p$. In particular, if $R$ is finite, then the set of functions mapping $R$ into itself is finite, whereas $R[x]$ is infinite, so some functions must be associated with infinitely many different polynomials. (In fact, it follows from the theory of cosets, applied to the additive groups involved, that every function that is associated with a polynomial must e associated with infinitely many polynomials.)

For example, if $R$ is the ring of integers modulo $p$, for $p$ a prime, then Fermat's Little Theorem states that the polynomials $x^p$ and $x$ are associated with the same functions mapping $R$ into itself.

Nevertheless, in many infinite rings (such as the ring of integers), this association of polynomials with functions is unique. In such contexts, the polynomials are often identified with their functions, by abuse of language. The association of polynomials with functions is an important one: polynomials were first studied as polynomial functions, and indeed it was not until recently that functions gained their modern definition, quite divorced from polynomials.

Finitude of Degree

"Polynomials of infinite degree" are properly called formal power series. The set of formal power series over a ring $R$ constitutes a ring, denoted $R[[x]]$, of which the ring of polynomials is a subring. In general, formal power series are not associated with mappings of $R$ into itself, as infinitely iterated addition is not generally well-defined.

See also

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