33:
157:
states that the product of primitive polynomials (with coefficients in the same unique factorization domain) also is primitive. This implies that the content and the primitive part of the product of two polynomials are, respectively, the product of the contents and the product of the primitive parts.
1899:
For factoring a multivariate polynomial over a field or over the integers, one may consider it as a univariate polynomial with coefficients in a polynomial ring with one less indeterminate. Then the factorization is reduced to factorizing separately the primitive part and the content. As the content
1542:
A consequence is that factoring polynomials over the rationals is equivalent to factoring primitive polynomials over the integers. As polynomials with coefficients in a field are more common than polynomials with integer coefficients, it may seem that this equivalence may be used for factoring
1228:
The last property implies that the computation of the primitive-part-content factorization of a polynomial reduces the computation of its complete factorization to the separate factorization of the content and the primitive part. This is generally interesting, because the computation of the
1204:
939:
1067:
1080:
1543:
polynomials with integer coefficients. In fact, the truth is exactly the opposite: every known efficient algorithm for factoring polynomials with rational coefficients uses this equivalence for reducing the problem
1467:
350:
550:
450:
1838:
838:
1526:
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845:
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746:, which asserts that the product of two primitive polynomials is primitive, where a polynomial is primitive if 1 is the greatest common divisor of its coefficients. This implies:
1304:
128:
of such a polynomial is the quotient of the polynomial by its content. Thus a polynomial is the product of its primitive part and its content, and this factorization is unique
253:
1748:
1904:. For factorizing the primitive part, the standard method consists of substituting integers to the indeterminates of the coefficients in a way that does not change the
62:
1472:
It is easy to show that this definition does not depend on the choice of the common denominator, and that the primitive-part-content factorization remains valid:
255:
may be either 2 or −2, since 2 is the greatest common divisor of −12, 30, and −20. If one chooses 2 as the content, the primitive part of this polynomial is
1632:
is a unique factorization domain. The same is true for a polynomial ring over a unique factorization domain. To prove this, it suffices to consider the
1199:{\displaystyle \operatorname {pp} (\operatorname {gcd} (P_{1},P_{2}))=\operatorname {gcd} (\operatorname {pp} (P_{1}),\operatorname {pp} (P_{2})).}
1571:
of many fractions, and this makes the
Euclidean algorithm less efficient than algorithms which work only with polynomials over the integers (see
1965:
1908:
in the remaining variable, factorizing the resulting univariate polynomial, and lifting the result to a factorization of the primitive part.
165:, the first step of a polynomial factorization algorithm is generally the computation of its primitive part–content factorization (see
1572:
181:
753:
2003:
1942:
1419:
84:
683:
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is defined for polynomials with rational coefficients. In fact, in this case, the
Euclidean algorithm requires one to compute the
261:
455:
For aesthetic reasons, one often prefers choosing a negative content, here −2, giving the primitive-part-content factorization
154:
743:
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361:
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45:
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1651:
divides a product, then it divides one of the factors. For univariate polynomials over a field, this results from
1478:
630:
is the greatest common divisor of its coefficients, and, as such, is defined up to multiplication by a unit. The
66:
1243:
The primitive-part-content factorization may be extended to polynomials with rational coefficients as follows.
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1341:
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193:
121:
1905:
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680:, then the primitive part must be changed by dividing it by the same unit, in order to keep the equality
169:). Then the factorization problem is reduced to factorize separately the content and the primitive part.
1917:
674:, which is unique up to multiplication by a unit. If the content is changed by multiplication by a unit
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2027:
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934:{\displaystyle \operatorname {pp} (P_{1}P_{2})=\operatorname {pp} (P_{1})\operatorname {pp} (P_{2}).}
210:
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This equivalence is also used for computing greatest common divisors of polynomials, although the
1071:
The primitive part of a greatest common divisor of polynomials is the greatest common divisor (in
17:
1616:
that a polynomial ring over a unique factorization domain is also a unique factorization domain.
1613:
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1544:
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184:
and factorization of polynomials over the integers and of polynomials over the rational numbers.
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200:. The choice is arbitrary, and may depend on a further convention, which is commonly that the
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of a unique factorization domain. This makes essentially equivalent the problems of computing
1062:{\displaystyle c(\operatorname {gcd} (P_{1},P_{2}))=\operatorname {gcd} (c(P_{1}),c(P_{2})).}
2022:
1971:
197:
151:
if its content equals 1. Thus the primitive part of a polynomial is a primitive polynomial.
943:
The content of a greatest common divisor of polynomials is the greatest common divisor (in
1975:
1625:
574:
173:
1224:) of the content and of the factorization (in the polynomial ring) of the primitive part.
842:
The primitive part of a product of polynomials is the product of their primitive parts:
1984:
1229:
prime-part-content factorization involves only greatest common divisor computation in
2016:
1930:
1861:, it is a primitive polynomial (because it is irreducible). Then Euclid's lemma in
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113:
1953:
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106:
161:
As the computation of greatest common divisors is generally much easier than
750:
The content of a product of polynomials is the product of their contents:
192:
For a polynomial with integer coefficients, the content may be either the
742:
The main properties of the content and the primitive part are results of
140:
of the coefficients (and the multiplication of the primitive part by the
1900:
has one less indeterminate, it may be factorized by applying the method
1846:, it divides one of the contents, and therefore one of the polynomials.
1252:
with rational coefficients, by rewriting its coefficients with the same
167:
Factorization of polynomials § Primitive part–content factorization
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110:
98:
172:
Content and primitive part may be generalized to polynomials over the
1535:
with a unique primitive polynomial over the integers, and that the
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1583:
The results of the preceding section remain valid if the ring of
560:
In the remainder of this article, we consider polynomials over a
1462:{\displaystyle \operatorname {pp} (P)=\operatorname {pp} (Q).}
26:
1643:
The unique factorization property is a direct consequence of
1668:
be a unique factorization domain, which is not a field, and
1587:
and the field of rationals are respectively replaced by any
733:
which is called the primitive-part-content factorization of
345:{\displaystyle -6x^{3}+15x-10={\frac {-12x^{3}+30x-20}{2}},}
591:
are well defined, and are unique up to multiplication by a
668:
by its content; it is a polynomial with coefficients in
1531:
This shows that every polynomial over the rationals is
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allows the computation of this primitive polynomial.
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and thus the primitive-part-content factorization is
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1235:, which is usually much easier than factorization.
1983:
1960:(Third ed.), Reading, Mass.: Addison-Wesley,
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545:{\displaystyle -12x^{3}+30x-20=-2(6x^{3}-15x+10).}
544:
445:{\displaystyle -12x^{3}+30x-20=2(-6x^{3}+15x-10).}
444:
344:
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1620:Unique factorization property of polynomial rings
1750:of two polynomials, then it divides the content
54:but its sources remain unclear because it lacks
1833:{\displaystyle c(P_{1}P_{2})=c(P_{1})c(P_{2}).}
1313:is a polynomial with integer coefficients. The
833:{\displaystyle c(P_{1}P_{2})=c(P_{1})c(P_{2}).}
176:, and, more generally, to polynomials over the
1521:{\displaystyle P=c(P)\operatorname {pp} (P).}
726:{\displaystyle P=c(P)\operatorname {pp} (P),}
116:(or, more generally, with coefficients in a
8:
1636:case, as the general case may be deduced by
1867:results immediately from Euclid's lemma in
1889:Factorization of multivariate polynomials
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85:Learn how and when to remove this message
1698:or an irreducible primitive polynomial.
1218:is the product of the factorization (in
1387:{\displaystyle c(P)={\frac {c(Q)}{d}},}
132:the multiplication of the content by a
1608:This is typically used for factoring
569:, which can typically be the ring of
7:
1692:is either an irreducible element in
1674:the univariate polynomial ring over
204:of the primitive part be positive.
1573:Polynomial greatest common divisor
25:
18:Primitive polynomial (ring theory)
1935:Rings, modules and linear algebra
1640:on the number of indeterminates.
1299:{\displaystyle P={\frac {Q}{d}},}
1655:, which itself results from the
31:
248:{\displaystyle -12x^{3}+30x-20}
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1879:is the field of fractions of
155:Gauss's lemma for polynomials
1895:Factorization of polynomials
1558:Factorization of polynomials
1077:) of their primitive parts:
207:For example, the content of
1840:Thus, by Euclid's lemma in
1589:unique factorization domain
562:unique factorization domain
196:of the coefficients or its
118:unique factorization domain
2044:
1992:Cambridge University Press
1892:
1743:{\displaystyle P_{1}P_{2}}
1680:. An irreducible element
1579:Over a field of fractions
1407:is the primitive part of
124:of its coefficients. The
1610:multivariate polynomials
589:greatest common divisors
182:greatest common divisors
163:polynomial factorization
40:This article includes a
1986:Rings and factorization
194:greatest common divisor
122:greatest common divisor
69:more precise citations.
1933:; T.O. Hawkes (1970).
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1713:and divides a product
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1982:David Sharpe (1987).
1918:Rational root theorem
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1212:of a polynomial over
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624:with coefficients in
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1937:. Chapman and Hall.
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1657:Euclidean algorithm
1649:irreducible element
1565:Euclidean algorithm
1537:Euclidean algorithm
1323:is the quotient by
1246:Given a polynomial
202:leading coefficient
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1740:
1598:field of fractions
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1329:of the content of
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1254:common denominator
1239:Over the rationals
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178:field of fractions
42:list of references
1967:978-0-201-55540-0
1653:Bézout's identity
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188:Over the integers
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16:(Redirected from
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147:A polynomial is
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928:
920:
916:
909:
906:
898:
894:
887:
884:
881:
873:
869:
863:
859:
852:
849:
841:
827:
819:
815:
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800:
796:
789:
786:
778:
774:
768:
764:
757:
749:
748:
747:
745:
744:Gauss's lemma
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714:
708:
705:
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672:
666:
659:
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628:
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611:
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602:
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572:
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563:
555:
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339:
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321:
318:
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296:
293:
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258:
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205:
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199:
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179:
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170:
168:
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159:
156:
152:
150:
145:
143:
139:
135:
131:
127:
123:
119:
115:
112:
108:
105:of a nonzero
104:
100:
89:
86:
78:
75:December 2018
68:
64:
58:
57:
51:
47:
43:
38:
29:
28:
19:
1985:
1957:
1951:Page 181 of
1934:
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1623:
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1582:
1569:reduced form
1562:
1552:
1549:prime number
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1471:
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1325:
1319:
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1308:
1263:
1257:
1248:
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1231:
1227:
1220:
1214:
1073:
945:
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597:
583:
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454:
354:
206:
191:
171:
160:
153:
148:
146:
125:
114:coefficients
102:
96:
81:
72:
61:Please help
53:
2028:Polynomials
1994:. pp.
1954:Lang, Serge
1902:recursively
1335:, that is
67:introducing
2017:Categories
1976:0848.13001
1931:B. Hartley
1924:References
1893:See also:
1634:univariate
1612:, and for
1533:associated
556:Properties
107:polynomial
1638:induction
1504:
1445:
1427:
1172:
1147:
1138:
1097:
1088:
1007:
966:
910:
888:
853:
709:
522:−
500:−
491:−
466:−
431:−
406:−
391:−
366:−
328:−
303:−
291:−
266:−
240:−
215:−
149:primitive
120:) is the
1956:(1993),
1912:See also
1873:, where
1662:So, let
1647:: If an
1596:and its
1585:integers
1397:and the
571:integers
2023:Algebra
1958:Algebra
1855:is not
1628:over a
1614:proving
1315:content
606:content
577:over a
573:, or a
142:inverse
136:of the
111:integer
103:content
99:algebra
63:improve
2002:
1974:
1964:
1941:
1906:degree
1707:is in
1545:modulo
1309:where
101:, the
1996:68–69
1630:field
1556:(see
1547:some
581:. In
579:field
130:up to
109:with
48:, or
2000:ISBN
1962:ISBN
1939:ISBN
604:The
593:unit
138:ring
134:unit
1972:Zbl
1849:If
1701:If
1686:in
1575:).
1560:).
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1317:of
1267:as
1135:gcd
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1004:gcd
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662:of
641:of
635:pp(
595:of
97:In
2019::
1998:.
1990:.
1970:,
1885:.
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285:15
243:20
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2008:.
1947:.
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1191:)
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1166:,
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1038:(
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761:(
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712:(
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