 Factors - Maple Programming Help

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Factors

inert factors function

 Calling Sequence Factors(a, K)

Parameters

 a - multivariate polynomial K - optional specification for an algebraic extension

Description

 • The Factors function is a placeholder for representing the factorization of the multivariate polynomial a over U, a unique factorization domain. The construct Factors(a) produces a data structure of the form $\left[u,\left[\left[{f}_{1},{e}_{1}\right],\mathrm{...},\left[{f}_{n},{e}_{n}\right]\right]\right]$ such that $a=u{f}_{1}^{{e}_{1}}\cdots {f}_{n}^{{e}_{n}}$, where each f[i] is a primitive irreducible polynomial.
 • The difference between the Factors function and the Factor function is only the form of the result.  The Factor function, if defined, returns a Maple sum of products more suitable for interactive display and manipulation.
 • The call Factors(a) mod p computes the factorization of a over the integers modulo p, a prime integer. The polynomial a must have rational coefficients or coefficients over a finite field specified by RootOfs.
 • The call Factors(a, K) mod p computes the factorization over the finite field defined by K, an algebraic extension of the integers mod p where K is a RootOf.
 • The call modp1(Factors(a),p) computes the factorization of the polynomial a in the $\mathrm{modp1}$ representation modulo p a prime integer.
 • The call evala(Factors(a, K)) computes the factorization of the polynomial a over an algebraic number (or function) field defined by the extension K, which is specified as a RootOf or a set of RootOfs. The polynomial a must have algebraic number (or function) coefficients. The factors are monic for the ordering of the variables chosen by Maple.

Examples

 > Factors(2*x^2+6*x+6) mod 7;
 $\left[{2}{,}\left[\left[{x}{+}{6}{,}{1}\right]{,}\left[{x}{+}{4}{,}{1}\right]\right]\right]$ (1)
 > Factors(x^5+1) mod 2;
 $\left[{1}{,}\left[\left[{x}{+}{1}{,}{1}\right]{,}\left[{{x}}^{{4}}{+}{{x}}^{{3}}{+}{{x}}^{{2}}{+}{x}{+}{1}{,}{1}\right]\right]\right]$ (2)
 > alias(alpha=RootOf(x^2+x+1));
 ${\mathrm{\alpha }}$ (3)
 > Factors(x^5+1,alpha) mod 2;
 $\left[{1}{,}\left[\left[{\mathrm{\alpha }}{}{x}{+}{{x}}^{{2}}{+}{1}{,}{1}\right]{,}\left[{x}{+}{1}{,}{1}\right]{,}\left[{\mathrm{\alpha }}{}{x}{+}{{x}}^{{2}}{+}{x}{+}{1}{,}{1}\right]\right]\right]$ (4)
 > alias(sqrt2=RootOf(x^2-2)):
 > evala(Factors(2*x^2-1,sqrt2));
 $\left[{2}{,}\left[\left[{x}{+}\frac{{\mathrm{sqrt2}}}{{2}}{,}{1}\right]{,}\left[{x}{-}\frac{{\mathrm{sqrt2}}}{{2}}{,}{1}\right]\right]\right]$ (5)
 > alias(sqrtx=RootOf(y^2-x,y)):
 > evala(Factors(x*y^2-1,sqrtx));
 $\left[{x}{,}\left[\left[{y}{-}\frac{{\mathrm{sqrtx}}}{{x}}{,}{1}\right]{,}\left[{y}{+}\frac{{\mathrm{sqrtx}}}{{x}}{,}{1}\right]\right]\right]$ (6)
 > expand((x^3+y^5+2)*(x*y^2+3)) mod 7;
 ${x}{}{{y}}^{{7}}{+}{{x}}^{{4}}{}{{y}}^{{2}}{+}{3}{}{{y}}^{{5}}{+}{3}{}{{x}}^{{3}}{+}{2}{}{x}{}{{y}}^{{2}}{+}{6}$ (7)
 > Factors((7)) mod 7;
 $\left[{1}{,}\left[\left[{{y}}^{{5}}{+}{{x}}^{{3}}{+}{2}{,}{1}\right]{,}\left[{x}{}{{y}}^{{2}}{+}{3}{,}{1}\right]\right]\right]$ (8)
 > Factors(x^2+2*x*y+y^2+1+x+y,alpha) mod 5;
 $\left[{1}{,}\left[\left[{y}{+}{x}{+}{4}{}{\mathrm{\alpha }}{,}{1}\right]{,}\left[{y}{+}{x}{+}{\mathrm{\alpha }}{+}{1}{,}{1}\right]\right]\right]$ (9)
 > Factors(x^2*y+x*y^2+2*alpha*x*y+alpha*x^2+4*alpha*x+y+alpha) mod 5;
 $\left[{1}{,}\left[\left[{\mathrm{\alpha }}{}{x}{+}{x}{}{y}{+}{1}{,}{1}\right]{,}\left[{y}{+}{x}{+}{\mathrm{\alpha }}{,}{1}\right]\right]\right]$ (10)