Computational Commutative Algebra by Kreuzer and Robbiano PDF

Algebraic Geometry

By Kreuzer and Robbiano

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For elements z ∈ Z[i] . Hint: Proceed as for integers, searching for divisors in the set of all elements a + bi such that a2 + b2 divides ϕ(z). Tutorial 5: Squarefree Parts of Polynomials In this tutorial we shall explore how one can effectively compute the squarefree part of a univariate polynomial over certain fields. It will turn out that this seemingly innocent problem is in fact intrinsically related to the structure of the base field K . a) Let K be a field of characteristic p > 0 , and let ϕ : K → K be the map defined by ϕ(a) = ap .

And a clever list construction. Write a CoCoA program ListListToPoly(. ) which converts lists of lists back to polynomials in K[x, y]. Apply the programs ReprPoly(. ) and ListListToPoly(. ) to the polynomials f1 = x2 + 2xy + 3y 2 , f2 = y 2 − x4 , and f3 = 1 + x + y + x2 + y 2 + x4 + y 4 + x8 + y 8 . Write CoCoA-programs AddPoly(. ) and MultPoly(. ) which take two lists L1 , L2 representing polynomials in K[x, y] and compute the lists representing their sum and product, respectively. Check the correctness of your programs by converting f1 + f2 , f1 · f2 and f2 f3 + f13 into lists in two ways.

Fm−1 ), fm ) d) Let f = c p∈P pαp with a unit c ∈ R , with αp ∈ N , and with αp = 0 for all but finitely many p ∈ P be the decomposition of an element f ∈ R \ {0} into irreducible factors. Then the element pmin{1,αp } sqfree(f ) = p∈P is called a squarefree part of f . 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 , .

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