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Polynomial ring
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== Definition (univariate case)== Let {{math|''K''}} be a [[field (mathematics)|field]] or (more generally) a [[commutative ring]]. The '''polynomial ring''' in {{math|''X''}} over {{math|''K''}}, which is denoted {{math|''K''[''X'']}}, can be defined in several equivalent ways. One of them is to define {{math|''K''[''X'']}} as the set of expressions, called '''polynomials''' in {{math|''X''}}, of the form<ref>{{harvnb|Herstein|1975|p=153}}</ref> :<math>p = p_0 + p_1 X + p_2 X^2 + \cdots + p_{m - 1} X^{m - 1} + p_m X^m,</math> where {{math|''p''<sub>0</sub>, ''p''<sub>1</sub>, …, ''p''<sub>''m''</sub>}}, the '''coefficients''' of {{math|''p''}}, are elements of {{math|''K''}}, {{math|''p{{sub|m}}'' ≠ 0}} if {{math|''m'' > 0}}, and {{math|''X'', ''X''{{i sup|2}}, …,}} are symbols, which are considered as "powers" of {{math|''X''}}, and follow the usual rules of [[exponentiation]]: {{math|1=''X''{{i sup|0}} = 1}}, {{math|1=''X''{{i sup|1}} = ''X''}}, and <math> X^k\, X^l = X^{k+l}</math> for any [[nonnegative integer]]s {{math|''k''}} and {{math|''l''}}. The symbol {{math|''X''}} is called an indeterminate<ref>Herstein, Hall p. 73</ref> or variable.<ref>{{harvnb|Lang|2002|p=97}}</ref> (The term of "variable" comes from the terminology of [[polynomial function]]s. However, here, {{mvar|X}} has no value (other than itself), and cannot vary, being a ''constant'' in the polynomial ring.) Two polynomials are equal when the corresponding coefficients of each {{math|''X''{{i sup|''k''}}}} are equal. One can think of the ring {{math|''K''[''X'']}} as arising from {{math|''K''}} by adding one new element {{math|''X''}} that is external to {{math|''K''}}, commutes with all elements of {{math|''K''}}, and has no other specific properties. This can be used for an equivalent definition of polynomial rings. The polynomial ring in {{math|''X''}} over {{math|''K''}} is equipped with an addition, a multiplication and a [[scalar multiplication]] that make it a [[Commutative algebra (structure)|commutative algebra]]. These operations are defined according to the ordinary rules for manipulating algebraic expressions. Specifically, if :<math>p = p_0 + p_1 X + p_2 X^2 + \cdots + p_m X^m,</math> and :<math>q = q_0 + q_1 X + q_2 X^2 + \cdots + q_n X^n,</math> then :<math>p + q = r_0 + r_1 X + r_2 X^2 + \cdots + r_k X^k,</math> and :<math>pq = s_0 + s_1 X + s_2 X^2 + \cdots + s_l X^l,</math> where {{math|1=''k'' = max(''m'', ''n''), ''l'' = ''m'' + ''n''}}, :<math>r_i = p_i + q_i</math> and :<math>s_i = p_0 q_i + p_1 q_{i-1} + \cdots + p_i q_0.</math> In these formulas, the polynomials {{math|''p''}} and {{math|''q''}} are extended by adding "dummy terms" with zero coefficients, so that all {{math|''p''<sub>''i''</sub>}} and {{math|''q''<sub>''i''</sub>}} that appear in the formulas are defined. Specifically, if {{math|''m'' < ''n''}}, then {{math|1=''p''<sub>''i''</sub> = 0}} for {{math|''m'' < ''i'' ≤ ''n''}}. The scalar multiplication is the special case of the multiplication where {{math|1=''p'' = ''p''<sub>0</sub>}} is reduced to its ''constant term'' (the term that is independent of {{math|''X''}}); that is :<math>p_0\left(q_0 + q_1 X + \dots + q_n X^n\right) = p_0 q_0 + \left(p_0 q_1\right)X + \cdots + \left(p_0 q_n\right)X^n</math> It is straightforward to verify that these three operations satisfy the axioms of a commutative algebra over {{mvar|K}}. Therefore, polynomial rings are also called ''polynomial algebras''. Another equivalent definition is often preferred, although less intuitive, because it is easier to make it completely rigorous, which consists in defining a polynomial as an infinite [[sequence]] {{math|(''p''<sub>0</sub>, ''p''<sub>1</sub>, ''p''<sub>2</sub>, …)}} of elements of {{math|''K''}}, having the property that only a finite number of the elements are nonzero, or equivalently, a sequence for which there is some {{math|''m''}} so that {{nowrap|1=''p''<sub>''n''</sub> = 0}} for {{math|''n'' > ''m''}}. In this case, {{math|''p''{{sub|0}}}} and {{mvar|X}} are considered as alternate notations for the sequences {{math|(''p''{{sub|0}}, 0, 0, …)}} and {{math|(0, 1, 0, 0, …)}}, respectively. A straightforward use of the operation rules shows that the expression :<math>p_0 + p_1 X + p_2 X^2 + \cdots + p_m X^m</math> is then an alternate notation for the sequence :{{math|(''p''<sub>0</sub>, ''p''<sub>1</sub>, ''p''<sub>2</sub>, …, ''p''<sub>''m''</sub>, 0, 0, …)}}. ===Terminology=== Let :<math>p = p_0 + p_1 X + p_2 X^2 + \cdots + p_{m - 1} X^{m - 1} + p_m X^m,</math> be a nonzero polynomial with <math>p_m\ne 0</math> The ''constant term'' of {{math|''p''}} is <math>p_0.</math> It is zero in the case of the zero polynomial. The ''degree'' of {{math|''p''}}, written {{math|deg(''p'')}} is <math>m,</math> the largest {{math|''k''}} such that the coefficient of {{math|''X''{{sup|''k''}}}} is not zero.<ref>{{harvnb|Herstein|1975|p=154}}</ref> The ''leading coefficient'' of {{math|''p''}} is <math>p_m.</math><ref>{{harvnb|Lang|2002|p=100}}</ref> In the special case of the zero polynomial, all of whose coefficients are zero, the leading coefficient is undefined, and 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=Wiley |year=2012|isbn=9780470647707|page=31|url=https://books.google.com/books?id=U2uv84cpJHQC&pg=RA1-PA31}}.</ref> defined to be {{math|−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=https://books.google.com/books?id=puWxs7KG2D0C&pg=PA250}}.</ref> or defined to be a {{math|−∞}}.<ref>{{citation|title=Elementary Matrix Theory|publisher=Dover|first=Howard Whitley|last=Eves|author-link=Howard Eves|year=1980|isbn=9780486150277|page=183|url=https://books.google.com/books?id=ayVxeUNbZRAC&pg=PA183}}.</ref> A ''constant polynomial'' is either the zero polynomial, or a polynomial of degree zero. A nonzero polynomial is [[monic polynomial|monic]] if its leading coefficient is <math>1.</math> Given two polynomials {{mvar|p}} and {{mvar|q}}, if the degree of the zero polynomial is defined to be <math>-\infty,</math> one has :<math>\deg(p+q) \le \max (\deg(p), \deg (q)),</math> and, over a [[field (mathematics)|field]], or more generally an [[integral domain]],<ref>{{harvnb|Herstein|1975|pp=155,162}}</ref> :<math>\deg(pq) = \deg(p) + \deg(q).</math> It follows immediately that, if {{math|''K''}} is an integral domain, then so is {{math|''K''[''X'']}}.<ref>{{harvnb|Herstein|1975|p=162}}</ref> It follows also that, if {{math|''K''}} is an integral domain, a polynomial is a [[unit (ring theory)|unit]] (that is, it has a [[multiplicative inverse]]) if and only if it is constant and is a unit in {{mvar|K}}. Two polynomials are [[associated element|associated]] if either one is the product of the other by a unit. Over a field, every nonzero polynomial is associated to a unique monic polynomial. Given two polynomials, {{mvar|p}} and {{mvar|q}}, one says that {{mvar|p}} ''divides'' {{mvar|q}}, {{mvar|p}} is a ''divisor'' of {{mvar|q}}, or {{mvar|q}} is a multiple of {{mvar|p}}, if there is a polynomial {{mvar|r}} such that {{math|1=''q'' = ''pr''}}. A polynomial is [[irreducible polynomial|irreducible]] if it is not the product of two non-constant polynomials, or equivalently, if its divisors are either constant polynomials or have the same degree. === Polynomial evaluation === {{further|Polynomial evaluation}} Let {{mvar|K}} be a field or, more generally, a [[commutative ring]], and {{mvar|R}} a ring containing {{mvar|K}}. For any polynomial {{mvar|P}} in {{math|''K''[''X'']}} and any element {{mvar|a}} in {{mvar|R}}, the substitution of {{mvar|X}} with {{mvar|a}} in {{mvar|P}} defines an element of {{math|''R''}}, which is [[Polynomial notation|denoted]] {{math|''P''(''a'')}}. This element is obtained by carrying on in {{mvar|R}} after the substitution the operations indicated by the expression of the polynomial. This computation is called the '''evaluation''' of {{math|''P''}} at {{math|''a''}}. For example, if we have :<math>P = X^2 - 1,</math> we have :<math>\begin{align} P(3) &= 3^2-1 = 8, \\ P(X^2+1) &= \left(X^2 + 1\right)^2 - 1 = X^4 + 2X^2 \end{align}</math> (in the first example {{math|1=''R'' = ''K''}}, and in the second one {{math|1=''R'' = ''K''[''X'']}}). Substituting {{math|''X''}} for itself results in :<math>P = P(X),</math> explaining why the sentences "Let {{mvar|P}} be a polynomial" and "Let {{math|''P''(''X'')}} be a polynomial" are equivalent. The ''polynomial function'' defined by a polynomial {{mvar|P}} is the function from {{mvar|K}} into {{mvar|K}} that is defined by <math>x\mapsto P(x).</math> If {{mvar|K}} is an infinite field, two different polynomials define different polynomial functions, but this property is false for finite fields. For example, if {{mvar|K}} is a field with {{mvar|q}} elements, then the polynomials {{math|0}} and {{math|''X''<sup>''q''</sup> − ''X''}} both define the zero function. For every {{math|''a''}} in {{math|''R''}}, the evaluation at {{mvar|a}}, that is, the map <math>P \mapsto P(a)</math> defines an [[algebra homomorphism]] from {{math|''K''[''X'']}} to {{math|''R''}}, which is the unique homomorphism from {{math|''K''[''X'']}} to {{math|''R''}} that fixes {{mvar|K}}, and maps {{mvar|X}} to {{mvar|a}}. In other words, {{math|''K''[''X'']}} has the following [[universal property]]: :For every ring {{mvar|R}} containing {{mvar|K}}, and every element {{mvar|a}} of {{mvar|R}}, there is a unique algebra homomorphism from {{math|''K''[''X'']}} to {{mvar|R}} that fixes {{mvar|K}}, and maps {{mvar|X}} to {{mvar|a}}. As for all universal properties, this defines the pair {{math|(''K''[''X''], ''X'')}} up to a unique isomorphism, and can therefore be taken as a definition of {{math|''K''[''X'']}}. The [[Image (mathematics)|image]] of the map <math>P \mapsto P(a)</math>, that is, the subset of {{mvar|R}} obtained by substituting {{mvar|a}} for {{mvar|X}} in elements of {{math|''K''[''X'']}}, is denoted {{math|''K''[''a'']}}.<ref>Knapp, Anthony W. (2006), ''Basic Algebra'', [[Birkhäuser]], p. 121.</ref> For example, <math>\Z[\sqrt{2}]=\{P(\sqrt{2})\mid P(X)\in\Z[X]\}</math>, and the simplification rules for the powers of a square root imply <math>\Z[\sqrt{2}]= \{a+b\sqrt 2 \mid a\in \Z, b\in \Z\}.</math>
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