The computation of polynomial greatest common divisors over an algebraic number field

Lars Langemyr, Scott McCallum
1989 Journal of symbolic computation  
We present a modular algorithm for computing the greatest common divisor of two polynomials over an algebraic number field. Our algorithm is an application of ideas of Brown and Collins. We use the Weinberge~-Rothschild homomorphic scheme with the important chaaage that we avoid factorinl~ the modular image of the minimal polynomial. We perform a computing time analysis and report, some empirical computing times. I n t r o d u c t i o n Algebraic number algorithms tend to be slow. Euclid's
more » ... ithm for computing polynomial greatest common divisors (god's) over an algebraic number field is no exception. Even for polynomials over the field Q of rational numbers, Euclid's algorithm can incur explosive coefficient growth. A modular algorithm independently developed by Brown (1971) and Collins (1972) eliminates tile problem of coefficient growth in polynomial gcd computation over the rational integers Z. In the present paper we develop a modular algorithm for polynomial gcd's over an algebraic number field using the basic ideas of Brown and Collins. Let a be a real algebraic number and let r be the primitive minimal polyrtomial of (~ over Z. Rubald (1974) gave algorithms for arithmetic in the field Q(cr). tie also discussed computing gcd's in Q(ce) [x]. He described how a generalization of the Brown-Collins method can be used when the following assumptions are satisfied: *Supported by STU mad NSEt~C.
doi:10.1016/s0747-7171(89)80053-7 fatcat:zvignluswjfb5bqa3efsbiha3q