h.fact

来自「强大的数学工具包」· FACT 代码 · 共 97 行

                               * * * * * * *
                               *    fact   *
                               * * * * * * *

        "factorization"
        SYNTAX: X=fact(A)

        A is an expression whose value is a nonzero element of Z, Q
        or a quadratic number field, a polynomial over Z, Q, Z/pZ 
        (where p is a prime) or over number fields or a rational 
        function or a univariate polynomial over GF(p^n).

        If A is an integer or a polynomial, fact computes the
        factorization of A. If A is a rational number or a rational
        function, fact computes the factorization of the numerator
        and denominator of A. If A is an algebraic number, fact
        computes the prime ideal factorization of the principal ideal
        (A).

        You can use nfon in order to factorize elements of Z, Q and
        polynomials over Z or Q over the number field, specified by
        curnf. You must consider the restrictions for factorization
        over number fields.
        
        Ideals in quadratic number fields are represented in the
        following form:

        If the number field is specified by x^2 +/- D, with D 
        squarefree:

        ((a)[b/s,(c+D^(1/2))/s]) = (a) * ((b/s)*Z + ((c+D^(1/2))/s)*Z)

        where s = 1 if D = 2 or 3 mod 4, s = 2 if D = 1 mod 4,
              a in N,
              b = min{ n in N : n is in (b/s*Z + (c+D^(1/2))/s*Z) },
              0 <= c < b.

        If the number field is specified by Ax^2+Bx+C, with A not 0:
        
        x0 is a root of the polynomial, h = B/A, k = C/A, 
        Dm^2 = h^2-4k with a rational number m and a squarefree
        integer D. Then the ideal in Q(x0)=Q(D^(1/2)) is represented as

        ((a)[b/ms,(c'+ 2x0)/ms]) = (a) * ((b/ms)*Z + ((c'+2x0)/ms)*Z)
  
        where c' = mc+h and s,a,b,c are the same as in the case
        of Q(D^(1/2)).

        See also A.J.Stephens and H.C.Williams: "Some Computational
                 Results on a Problem Concerning Powerful Numbers",
                 Math. Comp. V. 50, #182 (April 1988), pp 619-632.

        If A is an integer or a polynomial, X is the largest 
        prime factor of A. If A is a rational number or a rational
        function, X is the largest prime factor of the numerator
        of A. If A is an algebraic number, X is assigned the value
        of A.

        If A is an element of Q and its factorization is of the form:
        +- p1^e1 * ... * pN^eN, then AV[0]=p1, AV[1]=e1, ..., 
        AV[2*N-2]=pN, AV[2*N-1]=eN.
        If A is polynomial or a rational function with factorization
        c * p1^e1 * ... * pN^eN, then AV[0]=c, AV[1]=p1, AV[2]=e1, ...,
        AV[2*N-1]=pN, AV[2*N]=eN. (See "?avfunc".)

        Warning: If p > 2^30, the primality of p is not tested.


        Example 1: (correct)

                fact(36471289470)


        Example 2: (correct)

                fact(-360/49)


        Example 3: (correct)

                fact(NF(3 * Y - 2))


        Example 4: (correct)

                 fact(1/9 * x^2 + 1/7*x)
 

        Example 5: (correct)

                fact(x * y + x^2 * y^2)


        Example 6: (correct)

                fact((x^3 + x^2 + x) / (x^2 + 2*x +1))