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Theorem List for Metamath Proof Explorer - 20001-20100   *Has distinct variable group(s)
TypeLabelDescription
Statement

13.1.3  The division algorithm for univariate polynomials

Syntaxcmn1 20001 Monic polynomials.
Monic1p

Syntaxcuc1p 20002 Unitic polynomials.
Unic1p

Syntaxcq1p 20003 Univariate polynomial quotient.
quot1p

Syntaxcr1p 20004 Univariate polynomial remainder.
rem1p

Syntaxcig1p 20005 Univariate polynomial ideal generator.
idlGen1p

Definitiondf-mon1 20006* Define the set of monic univariate polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Monic1p Poly1 Poly1 coe1 deg1

Definitiondf-uc1p 20007* Define the set of unitic univariate polynomials, as the polynomials with an invertible leading coefficient. This is not a standard concept but is useful to us as the set of polynomials which can be used as the divisor in the polynomial division theorem ply1divalg 20013. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Unic1p Poly1 Poly1 coe1 deg1 Unit

Definitiondf-q1p 20008* Define the quotient of two univariate polynomials, which is guaranteed to exist and be unique by ply1divalg 20013. We actually use the reversed version for better harmony with our divisibility df-dvdsr 15701. (Contributed by Stefan O'Rear, 28-Mar-2015.)
quot1p Poly1 deg1 deg1

Definitiondf-r1p 20009* Define the remainder after dividing two univariate polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
rem1p Poly1 Poly1quot1pPoly1

Definitiondf-ig1p 20010* Define a choice function for generators of ideals over a division ring; this is the unique monic polynomial of minimal degree in the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.)
idlGen1p LIdealPoly1 Poly1 Poly1 Monic1p deg1 deg1 Poly1

Theoremply1divmo 20011* Uniqueness of a quotient in a polynomial division. For polynomials such that and the leading coefficient of is not a zero divisor, there is at most one polynomial which satisfies where the degree of is less than the degree of . (Contributed by Stefan O'Rear, 26-Mar-2015.) (Revised by NM, 17-Jun-2017.)
Poly1       deg1                                                                coe1        RLReg

Theoremply1divex 20012* Lemma for ply1divalg 20013: existence part. (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1       deg1                                                                                            coe1

Theoremply1divalg 20013* The division algorithm for univariate polynomials over a ring. For polynomials such that and the leading coefficient of is a unit, there are unique polynomials and such that the degree of is less than the degree of . (Contributed by Stefan O'Rear, 27-Mar-2015.)
Poly1       deg1                                                                coe1        Unit

Theoremply1divalg2 20014* Reverse the order of multiplication in ply1divalg 20013 via the opposite ring. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1       deg1                                                                coe1        Unit

Theoremuc1pval 20015* Value of the set of unitic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1                     deg1        Unic1p       Unit       coe1

Theoremisuc1p 20016 Being a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1                     deg1        Unic1p       Unit       coe1

Theoremmon1pval 20017* Value of the set of monic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1                     deg1        Monic1p              coe1

Theoremismon1p 20018 Being a monic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1                     deg1        Monic1p              coe1

Theoremuc1pcl 20019 Unitic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1              Unic1p

Theoremmon1pcl 20020 Monic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1              Monic1p

Theoremuc1pn0 20021 Unitic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1              Unic1p

Theoremmon1pn0 20022 Monic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1              Monic1p

Theoremuc1pdeg 20023 Unitic polynomials have nonnegative degrees. (Contributed by Stefan O'Rear, 28-Mar-2015.)
deg1        Unic1p

Theoremuc1pldg 20024 Unitic polynomials have unit leading coefficients. (Contributed by Stefan O'Rear, 28-Mar-2015.)
deg1        Unit       Unic1p       coe1

Theoremmon1pldg 20025 Unitic polynomials have one leading coefficients. (Contributed by Stefan O'Rear, 28-Mar-2015.)
deg1               Monic1p       coe1

Theoremmon1puc1p 20026 Monic polynomials are unitic. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Unic1p       Monic1p

Theoremuc1pmon1p 20027 Make a unitic polynomial monic by multiplying a factor to normalize the leading coefficient. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Unic1p       Monic1p       Poly1              algSc       deg1               coe1

Theoremdeg1submon1p 20028 The difference of two monic polynomials of the same degree is a polynomial of lesser degree. (Contributed by Stefan O'Rear, 28-Mar-2015.)
deg1        Monic1p       Poly1

Theoremq1pval 20029* Value of the univariate polynomial quotient function. (Contributed by Stefan O'Rear, 28-Mar-2015.)
quot1p       Poly1              deg1

Theoremq1peqb 20030 Characterizing property of the polynomial quotient. (Contributed by Stefan O'Rear, 28-Mar-2015.)
quot1p       Poly1              deg1                      Unic1p

Theoremq1pcl 20031 Closure of the quotient by a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
quot1p       Poly1              Unic1p

Theoremr1pval 20032 Value of the polynomial remainder function. (Contributed by Stefan O'Rear, 28-Mar-2015.)
rem1p       Poly1              quot1p

Theoremr1pcl 20033 Closure of remainder following division by a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)
rem1p       Poly1              Unic1p

Theoremr1pdeglt 20034 The remainder has a degree smaller than the divisor. (Contributed by Stefan O'Rear, 28-Mar-2015.)
rem1p       Poly1              Unic1p       deg1

Theoremr1pid 20035 Express the original polynomial as using the quotient and remainder functions for and . (Contributed by Mario Carneiro, 5-Jun-2015.)
Poly1              Unic1p       quot1p       rem1p

Theoremdvdsq1p 20036 Divisibility in a polynomial ring is witnessed by the quotient. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1       r              Unic1p              quot1p

Theoremdvdsr1p 20037 Divisibility in a polynomial ring in terms of the remainder. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Poly1       r              Unic1p              rem1p

Theoremply1remlem 20038 A term of the form is linear, monic, and has exactly one zero. (Contributed by Mario Carneiro, 12-Jun-2015.)
Poly1                     var1              algSc              eval1       NzRing                     Monic1p       deg1

Theoremply1rem 20039 The polynomial remainder theorem, or little Bézout's theorem (by contrast to the regular Bézout's theorem bezout 12997). If a polynomial is divided by the linear factor , the remainder is equal to , the evaluation of the polynomial at (interpreted as a constant polynomial). (Contributed by Mario Carneiro, 12-Jun-2015.)
Poly1                     var1              algSc              eval1       NzRing                            rem1p

Theoremfacth1 20040 The factor theorem and its converse. A polynomial has a root at iff is a factor of . (Contributed by Mario Carneiro, 12-Jun-2015.)
Poly1                     var1              algSc              eval1       NzRing                                   r

Theoremfta1glem1 20041 Lemma for fta1g 20043. (Contributed by Mario Carneiro, 7-Jun-2016.)
Poly1              deg1        eval1                     IDomn                     var1              algSc                                   quot1p

Theoremfta1glem2 20042* Lemma for fta1g 20043. (Contributed by Mario Carneiro, 12-Jun-2015.)
Poly1              deg1        eval1                     IDomn                     var1              algSc

Theoremfta1g 20043 The one-sided fundamental theorem of algebra. A polynomial of degree has at most roots. Unlike the real fundamental theorem fta 20815, which is only true in and other algebraically closed fields, this is true in any integral domain. (Contributed by Mario Carneiro, 12-Jun-2015.)
Poly1              deg1        eval1                     IDomn

Theoremfta1blem 20044 Lemma for fta1b 20045. (Contributed by Mario Carneiro, 14-Jun-2015.)
Poly1              deg1        eval1                                   var1

Theoremfta1b 20045* The assumption that be a domain in fta1g 20043 is necessary. Here we show that the statement is strong enough to prove that is a domain. (Contributed by Mario Carneiro, 12-Jun-2015.)
Poly1              deg1        eval1                     IDomn NzRing

Theoremdrnguc1p 20046 Over a division ring, all nonzero polynomials are unitic. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1                     Unic1p

Theoremig1peu 20047* There is a unique monic polynomial of minimal degree in any nonzero ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       LIdeal              Monic1p       deg1

Theoremig1pval 20048* Substitutions for the polynomial ideal generator function. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       idlGen1p              LIdeal       deg1        Monic1p

Theoremig1pval2 20049 Generator of the zero ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       idlGen1p

Theoremig1pval3 20050 Characterizing properties of the monic generator of a nonzero ideal of polynomials. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       idlGen1p              LIdeal       deg1        Monic1p

Theoremig1pcl 20051 The monic generator of an ideal is always in the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       idlGen1p       LIdeal

Theoremig1pdvds 20052 The monic generator of an ideal divides all elements of the ideal. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       idlGen1p       LIdeal       r

Theoremig1prsp 20053 Any ideal of polynomials over a division ring is generated by the ideal's canonical generator. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       idlGen1p       LIdeal       RSpan

Theoremply1lpir 20054 The ring of polynomials over a division ring has the principal ideal property. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       LPIR

Theoremply1pid 20055 The polynomials over a field are a PID. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Poly1       Field PID

13.1.4  Elementary properties of complex polynomials

Syntaxcply 20056 Extend class notation to include the set of complex polynomials.
Poly

Syntaxcidp 20057 Extend class notation to include the identity polynomial.

Syntaxccoe 20058 Extend class notation to include the coefficient function on polynomials.
coeff

Syntaxcdgr 20059 Extend class notation to include the degree function on polynomials.
deg

Definitiondf-ply 20060* Define the set of polynomials on the complexes with coefficients in the given subset. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Definitiondf-idp 20061 Define the identity polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)

Definitiondf-coe 20062* Define the coefficient function for a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff Poly

Definitiondf-dgr 20063 Define the degree of a polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
deg Poly coeff

Theoremplyco0 20064* Two ways to say that a function on the nonnegative integers has finite support. (Contributed by Mario Carneiro, 22-Jul-2014.)

Theoremplyval 20065* Value of the polynomial set function. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremplybss 20066 Reverse closure of the parameter of the polynomial set function. (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly

Theoremelply 20067* Definition of a polynomial with coefficients in . (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremelply2 20068* The coefficient function can be assumed to have zeroes outside . (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Poly

Theoremplyun0 20069 The set of polynomials is unaffected by the addition of zero. (This is built into the definition because all higher powers of a polynomial are effectively zero, so we require that the coefficient field contain zero to simplify some of our closure theorems.) (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly Poly

Theoremplyf 20070 The polynomial is a function on the complexes. (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly

Theoremplyss 20071 The polynomial set function preserves the subset relation. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly Poly

Theoremplyssc 20072 Every polynomial ring is contained in the ring of polynomials over . (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly Poly

Theoremelplyr 20073* Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Poly

Theoremelplyd 20074* Sufficient condition for elementhood in the set of polynomials. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremply1termlem 20075* Lemma for ply1term 20076. (Contributed by Mario Carneiro, 26-Jul-2014.)

Theoremply1term 20076* A one-term polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremplypow 20077* A power is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremplyconst 20078 A constant function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremne0p 20079 A test to show that a polynomial is nonzero. (Contributed by Mario Carneiro, 23-Jul-2014.)

Theoremply0 20080 The zero function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremplyid 20081 The identity function is a polynomial. (Contributed by Mario Carneiro, 17-Jul-2014.)
Poly

Theoremplyeq0lem 20082* Lemma for plyeq0 20083. If is the coefficient function for a nonzero polynomial such that for every and is the nonzero leading coefficient, then the function is a sum of powers of , and so the limit of this function as is the constant term, . But everywhere, so this limit is also equal to zero so that , a contradiction. (Contributed by Mario Carneiro, 22-Jul-2014.)

Theoremplyeq0 20083* If a polynomial is zero at every point (or even just zero at the positive integers), then all the coefficients must be zero. This is the basis for the method of equating coefficients of equal polynomials, and ensures that df-coe 20062 is well-defined. (Contributed by Mario Carneiro, 22-Jul-2014.)

Theoremplypf1 20084 Write the set of complex polynomials in a subring in terms of the abstract polynomial construction. (Contributed by Mario Carneiro, 3-Jul-2015.)
flds        Poly1              eval1fld       SubRingfld Poly

Theoremplyaddlem1 20085* Derive the coefficient function for the sum of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)
Poly       Poly

Theoremplymullem1 20086* Derive the coefficient function for the product of two polynomials. (Contributed by Mario Carneiro, 23-Jul-2014.)
Poly       Poly

Poly       Poly                                                                      Poly

Theoremplymullem 20088* Lemma for plymul 20090. (Contributed by Mario Carneiro, 21-Jul-2014.)
Poly       Poly                                                                             Poly

Theoremplyadd 20089* The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
Poly       Poly              Poly

Theoremplymul 20090* The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
Poly       Poly                     Poly

Theoremplysub 20091* The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.)
Poly       Poly                            Poly

Theoremplyaddcl 20092 The sum of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly Poly Poly

Theoremplymulcl 20093 The product of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly Poly Poly

Theoremplysubcl 20094 The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
Poly Poly Poly

Theoremcoeval 20095* Value of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly coeff

Theoremcoeeulem 20096* Lemma for coeeu 20097. (Contributed by Mario Carneiro, 22-Jul-2014.)
Poly

Theoremcoeeu 20097* Uniqueness of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Poly

Theoremcoelem 20098* Lemma for properties of the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Poly coeff coeff coeff

Theoremcoeeq 20099* If satisfies the properties of the coefficient function, it must be equal to the coefficient function. (Contributed by Mario Carneiro, 22-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Poly                                   coeff

Theoremdgrval 20100 Value of the degree function. (Contributed by Mario Carneiro, 22-Jul-2014.)
coeff       Poly deg

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