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:
$
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:
$
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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