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| | I'm Neil and I live in a seaside city in northern Austria, Nussberg. I'm 31 and I'm will soon finish my study at Environmental Studies.<br><br>Here is my blog ... [http://www.161994up.com http://www.161994up.com] |
| In [[mathematics]], especially in the field of [[abstract algebra]], a '''polynomial ring''' is a [[ring (mathematics)|ring]] formed from the [[Set (mathematics)|set]] of [[polynomial]]s in one or more [[indeterminate (variable)|indeterminate]]s (traditionally also called [[variable (mathematics)|variables]]) with coefficients in another [[ring (mathematics)|ring]], often a [[field (mathematics)|field]]. Polynomial rings have influenced much of mathematics, from the [[Hilbert basis theorem]], to the construction of [[splitting field]]s, and to the understanding of a [[linear operator]]. Many important conjectures involving polynomial rings, such as [[Quillen–Suslin theorem|Serre's problem]], have influenced the study of other rings, and have influenced even the definition of other rings, such as [[group ring]]s and [[formal power series|rings of formal power series]].
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| A closely related notion is that of the [[ring of polynomial functions]] on a [[vector space]].
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| == The polynomial ring ''K''[''X''] ==
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| === Definition ===
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| The '''polynomial ring''', ''K''[''X''], in ''X'' over a [[field (mathematics)|field]] ''K'' is defined<ref>Following Herstein p. 153</ref> as the set of expressions, called '''polynomials''' in ''X'', of the form
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| :<math>p = p_0 + p_1 X + p_2 X^2 + \cdots + p_{m - 1} X^{m - 1} + p_m X^m,</math>
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| where ''p''<sub>0</sub>, ''p''<sub>1</sub>,…, ''p''<sub>m</sub>, the '''coefficients''' of ''p'', are elements of ''K'', and ''X'', ''X'' <sup>2</sup>, are formal symbols ("the powers of ''X''"). By convention, ''X'' <sup>0</sup> = 1, ''X'' <sup>1</sup> = ''X'', and the product of the powers of ''X'' is defined by the familiar formula
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| :<math> X^k\, X^l = X^{k+l},</math>
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| where ''k'' and ''l'' are any [[natural number]]s. The symbol ''X'' is called an indeterminate<ref>Herstein, Hall p. 73</ref> or variable.<ref>Lang p. 97</ref>
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| Two polynomials are defined to be equal if and only if the corresponding coefficients for each power of ''X'' are equal, however terms with zero coefficient, 0''X'' <sup>''k''</sup>, may be added or omitted.
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| This terminology is suggested by [[real number|real]] or [[complex number|complex]] ''polynomial functions''. However, in general, ''X'' and its powers, ''X'' <sup>''k''</sup>, are treated as formal symbols, not as elements of the field ''K'' or functions over it. One can think of the ring ''K''[''X''] as arising from ''K'' by adding one new element ''X'' that is external to ''K'' and requiring that ''X'' commute with all elements of ''K''.
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| Polynomials in ''X'' are added and multiplied according to the ordinary rules for manipulating algebraic expressions, creating the structure of a ring. Specifically, if
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| :<math>p = p_0 + p_1 X + p_2 X^2 + \cdots + p_m X^m,</math>
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| and
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| :<math>q = q_0 + q_1 X + q_2 X^2 + \cdots + q_n X^n,</math>
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| then
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| :<math>p + q = r_0 + r_1 X + r_2 X^2 + \cdots + r_k X^k,</math>
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| and
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| :<math>pq = s_0 + s_1 X + s_2 X^2 + \cdots + s_l X^l,</math>
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| where
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| :<math>r_i=p_i+q_i</math>
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| and
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| :<math>s_i=p_0 q_i + p_1 q_{i-1} + \cdots + p_i q_0.</math>
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| If necessary, the polynomials ''p'' and ''q'' are extended by adding "dummy terms" with zero coefficients, so that the expressions for ''r''<sub>''i''</sub> and ''s''<sub>''i''</sub> are always defined.
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| A more rigorous, but less intuitive treatment<ref>Following Hall p.72-73</ref> defines a polynomial as an infinite [[tuple]], or ordered sequence of elements of ''K'', (''p''<sub>0</sub>, ''p''<sub>1</sub>, ''p''<sub>2</sub>, … ) having the property that only a finite number of the elements are nonzero, or equivalently, a sequence for which there is some ''m'' so that ''p''<sub>''n''</sub> = 0 for ''n''>''m''. In this case, the expression
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| :<math>p_0 + p_1 X + p_2 X^2 + \cdots + p_m X^m</math>
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| is considered an alternate notation for the sequence (''p''<sub>0</sub>, ''p''<sub>1</sub>, ''p''<sub>2</sub>, … ''p''<sub>''m''</sub>, 0, 0, …).
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| More generally, the field ''K'' can be replaced by any [[commutative ring]] ''R'' when taking the same construction as above, giving rise to the '''polynomial ring over''' ''R'', which is denoted ''R''[''X''].
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| ===Degree of a polynomial===
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| The '''degree''' of a polynomial ''p'', written deg(''p'') is the largest ''k'' such that the coefficient of ''X'' <sup>''k''</sup> is not zero.<ref>Herstein p. 154</ref> In this case the coefficient ''p''<sub>''k''</sub> is called the '''leading coefficient'''.<ref>Lang p.100</ref> In the special case of zero polynomial, all of whose coefficients are zero, the degree has been variously left undefined,<ref>{{citation|title=Calculus Single Variable|first1=Howard|last1=Anton|first2=Irl C.|last2=Bivens|first3=Stephen|last3=Davis|publisher=John Wiley & Sons|year=2012|isbn=9780470647707|page=31|url=http://books.google.com/books?id=U2uv84cpJHQC&pg=RA1-PA31}}.</ref> defined to be −1,<ref>{{citation|title=Rational Algebraic Curves: A Computer Algebra Approach|volume=22|series=Algorithms and Computation in Mathematics|first1=J. Rafael|last1=Sendra|first2=Franz|last2=Winkler|first3=Sonia|last3=Pérez-Diaz|publisher=Springer|year=2007|isbn=9783540737247|page=250|url=http://books.google.com/books?id=puWxs7KG2D0C&pg=PA250}}.</ref> or defined to be a special symbol −∞.<ref>{{citation|title=Elementary Matrix Theory|publisher=Dover|first=Howard Whitley|last=Eves|authorlink=Howard Eves|year=1980|isbn=9780486150277|page=183|url=http://books.google.com/books?id=ayVxeUNbZRAC&pg=PA183}}.</ref>
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| If ''K'' is a field, or more generally an [[integral domain]], then from the definition of multiplication,<ref>Herstein p.155, 162</ref>
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| :<math>\operatorname{deg}(pq) = \operatorname{deg}(p) + \operatorname{deg}(q).</math>
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| It follows immediately that if ''K'' is an integral domain then so is ''K''[''X''].<ref>Herstein p.162</ref>
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| === Properties of ''K''[''X''] ===
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| ==== Factorization in ''K''[''X''] ====
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| The next property of the polynomial ring is much deeper. Already [[Euclid]] noted that every positive integer can be uniquely factored into a product of [[prime number|primes]] — this statement is now called the [[fundamental theorem of arithmetic]]. The proof is based on [[Euclid's algorithm]] for finding the [[greatest common divisor]] of [[natural number]]s. At each step of this algorithm, a pair (''a'', ''b''), ''a'' > ''b'', of natural numbers is replaced by a new pair (''b'', ''r''), where ''r'' is the remainder from the division of ''a'' by ''b'', and the new numbers are ''smaller''. Gauss remarked that the procedure of division with the remainder can also be defined for polynomials: given two polynomials ''p'' and ''q'', where ''q'' ≠ 0, one can write
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| :<math> p = uq + r,\ </math>
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| where the quotient ''u'' and the remainder ''r'' are polynomials, the degree of ''r'' is less than the degree of ''q'', and a decomposition with these properties is unique. The quotient and the remainder are found using the [[polynomial long division]]. The degree of the polynomial now plays a role similar to the absolute value of an integer: it is strictly less in the remainder ''r'' than it is in ''q'', and when repeating this step such decrease cannot go on indefinitely. Therefore eventually some division will be exact, at which point the last non-zero remainder is the greatest common divisor of the initial two polynomials. Using the existence of greatest common divisors, Gauss was able to simultaneously rigorously prove the fundamental theorem of arithmetic for integers and its generalization to polynomials. In fact there exist other commutative rings than '''Z''' and ''K''[''X''] that similarly admit an analogue of the Euclidean algorithm; all such rings are called [[Euclidean ring]]s. Rings for which there exists unique (in an appropriate sense) factorization of nonzero elements into [[Irreducible element|irreducible]] factors are called ''[[unique factorization domain]]s'' or ''factorial rings''; the given construction shows that all Euclidean rings, and in particular '''Z''' and ''K''[''X''], are unique factorization domains.
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| Another corollary of the polynomial division with the remainder is the fact that every proper [[ideal (ring theory)|ideal]] ''I'' of ''K''[''X''] is [[principal ideal|principal]], i.e. ''I'' consists of the multiples of a single polynomial ''f''. Thus the polynomial ring ''K''[''X''] is a [[principal ideal domain]], and for the same reason every Euclidean domain is a principal ideal domain. Also every principal ideal domain is a unique-factorization domain. These deductions make essential use of the fact that the polynomial coefficients lie in a [[field (mathematics)|field]], namely in the polynomial division step, which requires the leading coefficient of ''q'', which is only known to be non-zero, to have an inverse. If ''R'' is an integral domain that is not a field then ''R''[''X''] is neither a Euclidean domain nor a principal ideal domain; however it could still be a unique factorization domain (and will be so if and only it ''R'' itself is a unique factorization domain, for instance if it is '''Z''' or another polynomial ring).
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| ==== Quotient ring of ''K''[''X''] ====
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| The ring ''K''[''X''] of polynomials over ''K'' is obtained from ''K'' by adjoining one element, ''X''. It turns out that any commutative ring ''L'' containing ''K'' and generated as a ring by a single element in addition to ''K'' can be described using ''K''[''X'']. In particular, this applies to finite [[field extension]]s of ''K''.
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| Suppose that a commutative ring ''L'' contains ''K'' and there exists an element ''θ'' of ''L'' such that the ring ''L'' is generated by ''θ'' over ''K''. Thus any element of ''L'' is a linear combination of powers of ''θ'' with coefficients in ''K''. Then there is a unique [[ring homomorphism]] ''φ'' from ''K''[''X''] into ''L'' which does not affect the elements of ''K'' itself (it is the [[identity function|identity map]] on ''K'') and maps each power of ''X'' to the same power of ''θ''. Its effect on the general polynomial amounts to "replacing ''X'' with ''θ''":
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| : <math> \varphi(a_m X^m + a_{m - 1} X^{m - 1} + \cdots + a_1 X + a_0) =
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| a_m \theta^m + a_{m - 1} \theta^{m - 1} + \cdots + a_1 \theta + a_0.</math>
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| By the assumption, any element of ''L'' appears as the right hand side of the last expression for suitable ''m'' and elements ''a''<sub>0</sub>, …, ''a''<sub>m</sub> of ''K''. Therefore, ''φ'' is [[surjective]] and ''L'' is a homomorphic image of ''K''[''X'']. More formally, let Ker ''φ'' be the [[kernel (algebra)|kernel]] of ''φ''. It is an [[ideal (ring theory)|ideal]] of ''K''[''X''] and by the first [[isomorphism theorem]] for rings, ''L'' is [[isomorphic]] to the quotient of the polynomial ring ''K''[''X''] by the ideal Ker ''φ''. Since the polynomial ring is a principal ideal domain, this ideal is [[principal ideal|principal]]: there exists a polynomial ''p''∈''K''[''X''] such that
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| : <math> L \simeq K[X]/(p). </math>
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| A particularly important application is to the case when the larger ring ''L'' is a [[field (mathematics)|field]]. Then the polynomial ''p'' must be [[irreducible polynomial|irreducible]]. Conversely, the [[primitive element theorem]] states that any finite separable field extension ''L''/''K'' can be generated by a single element ''θ''∈''L'' and the preceding theory then gives a concrete description of the field ''L'' as the quotient of the polynomial ring ''K''[''X''] by a principal ideal generated by an irreducible polynomial ''p''. As an illustration, the field '''C''' of [[complex number]]s is an extension of the field '''R''' of [[real number]]s generated by a single element ''i'' such that ''i''<sup>2</sup> + 1 = 0. Accordingly, the polynomial ''X''<sup>2</sup> + 1 is irreducible over '''R''' and
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| : <math> \mathbb{C} \simeq \mathbb{R}[X]/(X^2+1). </math>
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| More generally, given a (not necessarily commutative) ring ''A'' containing ''K'' and an element ''a'' of ''A'' that commutes with all elements of ''K'', there is a unique ring homomorphism from the polynomial ring ''K''[X] to ''A'' that maps ''X'' to ''a'':
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| :<math> \phi: K[X]\to A, \quad \phi(X)=a.</math>
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| This homomorphism is given by the same formula as before, but it is not surjective in general. The existence and uniqueness of such a homomorphism ''φ'' expresses a certain [[universal property]] of the ring of polynomials in one variable and explains ubiquity of polynomial rings in various questions and constructions of [[ring theory]] and [[commutative algebra]].
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| === Modules ===
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| The [[structure theorem for finitely generated modules over a principal ideal domain]] applies over
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| ''K''[''X''].. This means that every finitely generated module over ''K''[''X''] may be decomposed into a [[direct sum]] of a [[free module]] and finitely many modules of the form <math>K[X]/\langle P^k \rangle</math>, where ''P'' is an [[irreducible polynomial]] over ''K'' and ''k'' a positive integer.
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| == Polynomial evaluation ==
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| Let ''K'' be a field or, more generally, a [[commutative ring]], and ''R'' a ring containing ''K''. For any polynomial ''P'' in ''K''[X] and any element ''a'' in ''R'', the substitution of ''X'' by ''a'' in ''P'' defines an element of ''R'', which is denoted [[Polynomial notation|''P''(''a'')]]. This element is obtained by, after the substitution, carrying on, in ''R'', the operations indicated by the expression of the polynomial. This computation is called the ''evaluation'' of ''P'' at ''a''. For example, if we have
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| :<math>P=X^2-1,</math>
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| we have
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| :<math>P(3) = 3^2-1 = 8, </math>
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| :<math>P(X^2+1) = (X^2+1)^2-1 = X^4+2X^2</math>
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| (in the first example ''R'' = ''K'', and in the second one ''R'' = ''K''[''X'']). Substituting ''X'' by itself results in
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| :<math>P=P(X),</math>
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| explaining why the sentences "''Let P be a polynomial''" and "''Let P''(''X'') ''be a polynomial''" are equivalent.
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| For every ''a'' in ''R'', the map <math>P \mapsto P(a)</math> defines a [[ring homomorphism]] from ''K''[''X''] into ''R''.
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| The ''polynomial function'' defined by a polynomial ''P'' is the function from ''K'' into ''K'' that is defined by <math>x\mapsto P(x).</math> If ''K'' is an infinite field, two different polynomials define different polynomial functions, but this property is false for finite fields. For example, if ''K'' is a field with ''q'' elements, then the polynomials 0 and ''X''<sup>''q''</sup>-''X'' both define the zero function.
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| =={{Anchor|multivariable}}The polynomial ring in several variables==
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| === Polynomials ===
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| A polynomial in ''n'' variables ''X''<sub>1</sub>,…, ''X''<sub>''n''</sub> with coefficients in a field ''K'' is defined analogously to a polynomial in one variable, but the notation is more cumbersome. For any [[multi-index]] ''α'' = (''α''<sub>1</sub>,…, ''α''<sub>''n''</sub>), where each ''α''<sub>''i''</sub> is a non-negative integer, let
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| :<math> X^\alpha = \prod_{i=1}^n X_i^{\alpha_i} =
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| X_1^{\alpha_1}\ldots X_n^{\alpha_n}, \quad
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| p_\alpha = p_{\alpha_1\ldots\alpha_n}\in\mathbb{K}.\ </math>
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| The product ''X''<sup>''α''</sup> is called the '''[[monomial]]''' of multidegree ''α''. A '''polynomial''' is a finite linear combination of monomials with coefficients in ''K''
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| :<math> p = \sum_\alpha p_\alpha X^\alpha,\ </math>
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| and only finitely many coefficients ''p''<sub>''α''</sub> are different from 0. The '''degree''' of a monomial ''X''<sup>''α''</sup>, frequently denoted |''α''|, is defined as
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| :<math> |\alpha| = \sum_{i=1}^n \alpha_i,\ </math>
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| and the degree of a polynomial ''p'' is the largest degree of a monomial occurring with non-zero coefficient in the expansion of ''p''.
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| === The polynomial ring ===
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| Polynomials in ''n'' variables with coefficients in ''K'' form a commutative ring denoted
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| ''K''[''X''<sub>1</sub>,…, ''X''<sub>''n''</sub>], or sometimes ''K''[''X''], where ''X'' is a symbol representing the full set of variables, ''X'' = (''X''<sub>1</sub>,…, ''X''<sub>''n''</sub>), and called the '''polynomial ring in ''n'' variables'''. The polynomial ring in ''n'' variables can be obtained by repeated application of ''K''[''X''] (the order by which is irrelevant). For example, ''K''[''X''<sub>1</sub>, ''X''<sub>2</sub>] is [[Ring isomorphism|isomorphic]] to ''K''[''X''<sub>1</sub>][''X''<sub>2</sub>]. This ring plays fundamental role in [[algebraic geometry]]. Many results in [[commutative algebra|commutative]] and [[homological algebra]] originated in the study of its ideals and modules over this ring.
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| A polynomial ring with coefficients in <math>\mathbb{Z}</math> is the '''free commutative ring''' over its set of variables.
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| === Hilbert's Nullstellensatz ===
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| {{Main|Hilbert's Nullstellensatz}}
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| A group of fundamental results concerning the relation between ideals of the polynomial ring ''K''[''X''<sub>1</sub>,…, ''X''<sub>''n''</sub>] and [[algebraic set|algebraic subsets]] of ''K''<sup>''n''</sup> originating with [[David Hilbert]] is known under the name '''Nullstellensatz''' (literally: "zero-locus theorem").
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| * (''Weak form, algebraically closed field of coefficients''). Let ''K'' be an [[algebraically closed field]]. Then every [[maximal ideal]] ''m'' of ''K''[''X''<sub>1</sub>,…, ''X''<sub>''n''</sub>] has the form
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| :: <math> m = (X_1-a_1, \ldots, X_n-a_n),
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| \quad a = (a_1, \ldots, a_n) \in \mathbb{K}^n. </math>
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| * (''Weak form, any field of coefficients''). Let ''k'' be a field, ''K'' be an [[algebraically closed]] [[field extension]] of ''k'', and ''I'' be an ideal in the polynomial ring ''k''[''X''<sub>1</sub>,…, ''X''<sub>''n''</sub>]. Then ''I'' contains 1 if and only if the polynomials in ''I'' do not have any common zero in ''K''<sup>''n''</sup>.
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| * (''Strong form''). Let ''k'' be a field, ''K'' be an [[algebraically closed]] [[field extension]] of ''k'', ''I'' be an ideal in the polynomial ring ''k''[''X''<sub>1</sub>,…, ''X''<sub>''n''</sub>],and ''V''(''I'') be the algebraic subset of ''K''<sup>''n''</sup> defined by ''I''. Suppose that ''f'' is a polynomial which vanishes at all points of ''V''(''I''). Then some power of ''f'' belongs to the ideal ''I'':
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| :: <math> f^m \in I, \text{ for some } m\in \mathbb{N}. \, </math>
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| : Using the notion of the [[radical of an ideal]], the conclusion says that ''f'' belongs to the radical of ''I''. As a corollary of this form of Nullstellensatz, there is a [[bijective]] correspondence between the radical ideals of ''K''[''X''<sub>1</sub>,…, ''X''<sub>''n''</sub>] for an algebraically closed field ''K'' and the algebraic subsets of the ''n''-dimensional [[affine space]] ''K''<sup>''n''</sup>. It arises from the map
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| :: <math> I \mapsto V(I), \quad I\subset K[X_1,\ldots,X_n], \quad V(I)\subset K^n. </math>
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| : The [[prime ideal]]s of the polynomial ring correspond to [[irreducible component|irreducible]] subvarieties of ''K''<sup>''n''</sup>.
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| == Properties of the ring extension ''R'' ⊂ ''R''[''X''] ==
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| <!-- This is still in preliminary form -->
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| One of the basic techniques in [[commutative algebra]] is to relate properties of a ring with properties of its [[subring]]s. The notation ''R'' ⊂ ''S'' indicates that a ring ''R'' is a subring of a ring ''S''. In this case ''S'' is called an ''overring'' of ''R'' and one speaks of a '''ring extension'''. This works particularly well for polynomial rings and allows one to establish many important properties of the ring of polynomials in several variables over a field, ''K''[''X''<sub>1</sub>,…, ''X''<sub>''n''</sub>], by induction in ''n''.
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| === Summary of the results ===
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| In the following properties, ''R'' is a commutative ring and ''S'' = ''R''[''X''<sub>1</sub>,…, ''X''<sub>''n''</sub>] is the ring of polynomials in ''n'' variables over ''R''. The ring extension ''R'' ⊂ ''S'' can be built from ''R'' in ''n'' steps, by successively adjoining ''X''<sub>1</sub>,…, ''X''<sub>''n''</sub>. Thus to establish each of the properties below, it is sufficient to consider the case ''n'' = 1.
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| * If ''R'' is an [[integral domain]] then the same holds for ''S''.
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| * If ''R'' is a [[unique factorization domain]] then the same holds for ''S''. The proof is based on the [[Gauss's lemma (polynomial)|Gauss lemma]].
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| * '''[[Hilbert's basis theorem]]''': If ''R'' is a [[Noetherian ring]], then the same holds for ''S''.
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| * Suppose that ''R'' is a Noetherian ring of finite [[global dimension]]. Then
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| :: <math> \operatorname{gl}\,\dim R[X_1,\ldots,X_n] = \operatorname{gl}\, \dim R + n.</math>
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| : An analogous result holds for [[Krull dimension]].
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| ==Generalizations==
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| Polynomial rings have been generalized in a great many ways, including polynomial rings with generalized exponents, power series rings, [[noncommutative polynomial ring]]s, and skew-polynomial rings.
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| === Infinitely many variables ===
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| The possibility to allow an infinite set of indeterminates is not really a generalization, as the ordinary notion of polynomial ring allows for it. It is then still true that each monomial involves only a finite number of indeterminates (so that its degree remains finite), and that each polynomial is a linear combination of monomials, which by definition involves only finitely many of them. This explains why such polynomial rings are relatively seldom considered: each individual polynomial involves only finitely many indeterminates, and even any finite computation involving polynomials remains inside some subring of polynomials in finitely many indeterminates.
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| In the case of infinitely many indeterminates, one can consider a ring strictly larger than the polynomial ring but smaller than the power series ring, by taking the subring of the latter formed by power series whose monomials have a bounded degree. Its elements still have a finite degree and are therefore are somewhat like polynomials, but it is possible for instance to take the sum of all indeterminates, which is not a polynomial. A ring of this kind plays a role in constructing the [[ring of symmetric functions]].
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| ===Generalized exponents===
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| {{Main|Monoid ring}}
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| A simple generalization only changes the set from which the exponents on the variable are drawn. The formulas for addition and multiplication make sense as long as one can add exponents: ''X''<sup>''i''</sup>·''X''<sup>''j''</sup> = ''X''<sup>''i''+''j''</sup>. A set for which addition makes sense (is closed and associative) is called a [[monoid]]. The set of functions from a monoid ''N'' to a ring ''R'' which are nonzero at only finitely many places can be given the structure of a ring known as ''R''[''N''], the '''monoid ring''' of ''N'' with coefficients in ''R''. The addition is defined component-wise, so that if ''c'' = ''a''+''b'', then ''c''<sub>''n''</sub> = ''a''<sub>''n''</sub> + ''b''<sub>''n''</sub> for every ''n'' in ''N''. The multiplication is defined as the Cauchy product, so that if ''c'' = ''a''·''b'', then for each ''n'' in ''N'', ''c''<sub>''n''</sub> is the sum of all ''a''<sub>''i''</sub>''b''<sub>''j''</sub> where ''i'', ''j'' range over all pairs of elements of ''N'' which sum to ''n''.
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| When ''N'' is commutative, it is convenient to denote the function ''a'' in ''R''[''N''] as the formal sum:
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| :<math>\sum_{n \in N} a_n X^n</math>
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| and then the formulas for addition and multiplication are the familiar:
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| :<math>\left(\sum_{n \in N} a_n X^n\right) + \left(\sum_{n \in N} b_n X^n\right) = \sum_{n \in N} (a_n+b_n)X^n</math>
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| and
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| :<math>\left(\sum_{n \in N} a_n X^n\right) \cdot \left(\sum_{n \in N} b_n X^n\right) = \sum_{n \in N} \left( \sum_{i+j=n} a_ib_j\right)X^n</math>
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| where the latter sum is taken over all ''i'', ''j'' in ''N'' that sum to ''n''.
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| Some authors such as {{harv|Lang|2002|loc=II,§3}} go so far as to take this monoid definition as the starting point, and regular single variable polynomials are the special case where ''N'' is the monoid of non-negative integers. Polynomials in several variables simply take ''N'' to be the direct product of several copies of the monoid of non-negative integers. <!-- Quite tempting to say, ''N'' = '''N'''<sup>''n''</sup>. -->
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| Several interesting examples of rings and groups are formed by taking ''N'' to be the additive monoid of non-negative rational numbers, {{harv|Osbourne|2000|loc=§4.4}}.
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| ===Power series===
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| {{Main|Formal power series}}
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| Power series generalize the choice of exponent in a different direction by allowing infinitely many nonzero terms. This requires various hypotheses on the monoid ''N'' used for the exponents, to ensure that the sums in the Cauchy product are finite sums. Alternatively, a topology can be placed on the ring, and then one restricts to convergent infinite sums. For the standard choice of ''N'', the non-negative integers, there is no trouble, and the ring of formal power series is defined as the set of functions from ''N'' to a ring ''R'' with addition component-wise, and multiplication given by the Cauchy product. The ring of power series can be seen as the completion of the polynomial ring.
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| ===Noncommutative polynomial rings===
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| {{Main|Free algebra}}
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| For polynomial rings of more than one variable, the products ''X''·''Y'' and ''Y''·''X'' are simply defined to be equal. A more general notion of polynomial ring is obtained when the distinction between these two formal products is maintained. Formally, the polynomial ring in ''n'' noncommuting variables with coefficients in the ring ''R'' is the [[monoid ring]] ''R''[''N''], where the monoid ''N'' is the [[free monoid]] on ''n'' letters, also known as the set of all strings over an alphabet of ''n'' symbols, with multiplication given by concatenation. Neither the coefficients nor the variables need commute amongst themselves, but the coefficients and variables commute with each other.
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| Just as the polynomial ring in ''n'' variables with coefficients in the commutative ring ''R'' is the free commutative ''R''-algebra of rank ''n'', the noncommutative polynomial ring in ''n'' variables with coefficients in the commutative ring ''R'' is the free associative, unital ''R''-algebra on ''n'' generators, which is noncommutative when ''n'' > 1.
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| ===Differential and skew-polynomial rings===
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| {{Main|Ore extension}}
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| Other generalizations of polynomials are differential and skew-polynomial rings.
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| A '''differential polynomial ring''' is a ring of [[differential operator]]s formed from a ring ''R'' and a [[Derivation (abstract algebra)|derivation]] ''δ'' of ''R'' into ''R''. This derivation operates on ''R'', and will be denoted ''X'', when viewed as an operator. The elements of ''R'' also operate on ''R'' by multiplication. The [[function composition|composition of operators]] is denoted as the usual multiplication. It follows that the relation ''δ''(''ab'') = ''aδ''(''b'') + ''δ''(''a'')''b'' may be rewritten
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| :<math>X\cdot a = a\cdot X +\delta(a).</math>
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| This relation may be extended to define a skew multiplication between two polynomials in ''X'' with coefficients in ''R'', which make them a non-commutative ring.
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| The standard example, called a [[Weyl algebra]], takes ''R'' to be a (usual) polynomial ring ''k''[''Y''], and ''δ'' to be the standard polynomial derivative <math>\tfrac{\partial}{\partial Y}</math>. Taking ''a'' =''Y'' in the above relation, one gets the [[canonical commutation relation]], ''X''·''Y'' − ''Y''·''X'' = 1. Extending this relation by associativity and distributivity allows to construct explicitly the [[Weyl algebra]].{{harv|Lam|2001|loc=§1,ex1.9}}.
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| The '''skew-polynomial ring''' is defined similarly for a ring ''R'' and a ring endomorphism ''f'' of ''R'', by extending the multiplication from the relation ''X''·''r'' = ''f''(''r'')·''X'' to produce an associative multiplication that distributes over the standard addition. More generally, given a homomorphism ''F'' from the monoid '''N''' of the positive integers into the endomorphism ring of ''R'', the formula ''X''<sup>''n''</sup>·''r'' = ''F''(''n'')(''r'')·''X''<sup>''n''</sup> allows to construct a skew-polynomial ring.{{harv|Lam|2001|loc=§1,ex 1.11}} Skew polynomial rings are closely related to [[crossed product]] algebras.
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| == See also ==
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| * [[Additive polynomial]]
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| * [[Laurent polynomial]]
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| ==References==
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| <references/>
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| *{{cite book |title=An Introduction to Abstract Algebra|volume=2|first=F.M.|last=Hall|publisher=Cambridge University Press|year=1969|isbn=0521084849 |section=Section 3.6}}
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| *{{cite book |title=Topics in Algebra|first=I.N.|last=Herstein|publisher=Wiley |year=1975|isbn=0471010901|section=Section 3.9}}
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| *{{Citation | last1=Lam | first1=Tsit-Yuen | authorlink=Tsit Yuen Lam |title=A First Course in Noncommutative Rings | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-95325-0 | year=2001}}
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| *{{Lang Algebra}}
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| *{{Citation | last1=Osborne | first1=M. Scott | title=Basic homological algebra | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-98934-1 | id={{MathSciNet | id = 1757274}} | year=2000 | volume=196}}
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| [[Category:Commutative algebra]]
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| [[Category:Invariant theory]]
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| [[Category:Ring theory]]
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| [[Category:Polynomials]]
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| [[Category:Free algebraic structures]]
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| [[ja:多項式#多項式環]]
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