By B.H. Gross, B. Mazur

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Extra resources for Arithmetic on Elliptic Curves with Complex Multiplication

Example text

It is clear that the definition of greatest common divisors and least common multiples does not depend on the order of the elements. It is also clear that greatest common divisors, least common multiples, and squarefree parts of elements f1 , . . , fm ∈ R\{0} change only by a unit if we choose a different set of representatives P for the equivalence classes of irreducible elements. We shall therefore speak of the greatest common divisor and the least common multiple of f1 , . . , fm ∈ R\{0} , as well as the squarefree part of f ∈ R\{0}, while always keeping in mind that they are unique only up to a unit.

The elements in Supp(f ) , are written. Using the recursive definition of multivariate polynomials, we see that the way of writing them depends on how we write the univariate ones. And to do it, we see the necessity of knowing how to order T1 . Look again at f (x) = 1 + 2x − 3x2 , whose support is {1, x, x2 } . There are six ways of ordering three elements, which then yield six representations of f , namely 1 + 2x − 3x2 , 1 − 3x2 + 2x , 2x + 1 − 3x2 , 2x − 3x2 + 1, −3x2 + 1 + 2x , and −3x2 + 2x + 1.

Then check, if the representation f= µi j=1 κ∈K gcd(fi j , gi+1 − κ) consists of r different non-constant factors. If not, increase i by one and repeat step 3), until it does. Then return this representation as the result. Show that this is an algorithm which stops for some i ≤ r − 1 and that it returns a representation of f as the product of powers of distinct irreducible monic polynomials. It is called Berlekamp’s Algorithm. If we combine it with the algorithm for computing squarefree parts of polynomials in K[x] described in Tutorial 5, we have a complete factorization algorithm for univariate polynomials over finite fields.

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