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By Hans Sterk

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Extra resources for Algebra 3: algorithms in algebra [Lecture notes]

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V0 (θ) = ql+1 θl+1 +· · ·+q0 . Equating both expressions yields a system of equations in the qj . Depending on whether these equations admit a solution, the integral is elementary. Here is an example of how this works out. 18 Example. , of degree 1. So we start with x θ = q 2 θ 2 + q1 θ + q 0 , 52 Symbolic integration and obtain the system of equations q2 = 0, x = 2q2 θ + q1 , 0 = q 1 θ + q0 . For the moment we ignore the first equation (it says that q2 is a constant). Integrating the second equation gives 21 x2 +γ = 2q2 θ +q1 , with γ a constant, so that q2 = 0 and q1 = 21 x2 +γ.

The resultant of f, g is an element of k that determines if f and g have a common factor or not: f and g have a nonconstant common factor if and only if the resultant is 0. If f and g have a common factor h, then there is a polynomial relation of the form Af + Bg = 0 with deg(A) < deg(g) and deg(B) < deg(f ): simply take A = g/h and B = −f /h. The following lemma states that the converse also holds. 2 Lemma. Let f and g be of positive degree. Then f and g have a factor in common if and only if there exist nontrivial polynomials A and B satisfying deg(A) < deg(g) and deg(B) < deg(f ) such that Af + Bg = 0.

In the following it will be essential to work with polynomials with integer coefficients. Fortunately, factors of a polynomial with integer coefficients can always be taken in Z [X]. To explain this, we denote by c(f ) the gcd of the coefficients of f ∈ Z [X], the so–called content of f . The first statement in the following proposition is usually called the Lemma of Gauss. 2 Proposition. a) The content is multiplicative in the sense that for nonzero polynomials f, g ∈ Z [X] the relation c(f g) = c(f ) c(g) holds.

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