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14 Theorem. (Rothstein–Trager) Let K(x) be a differential field in one variable with field of constants K. Let a, b ∈ K[x] be relatively prime with deg(a) < deg(b) and with b monic. Then the minimal algebraic extension of K such that the integral of a/b can be expressed as a = b ci log(vi ), i with ci ∈ K and vi ∈ K[x], is the splitting field K of the resultant R(z) = resx (a − z D(b), b) ∈ K[z]. In fact, the ci are the roots of R(z) and vi = resx (a − ci D(b), b) for every i. Proof. First we determine the ci as roots of the resultant mentioned.

3 From the following section on, all fields in this chapter have characteristic zero. 1 Differential fields From the point of view of differentiation, polynomials and rational functions are quite simple. In the algebraic context, differentiation can be viewed as a certain operator on the field K(x) of rational functions over the field K. In this algebraic setting more complicated expressions, like the logarithm, can be treated by adjoining them to the field K(x). For this to make sense we need to be able to algebraically characterize such expressions and to define how differentiation extends to this larger field.

Altering the sign of a if necessary, we can arrange it so that ab = 1 and f = (ag) (bh). 3 Now an obvious operation on polynomials with integer coefficients is to reduce their coefficients modulo a prime. Let p be a prime and let f = am X m +am−1 X m−1 +· · ·+a1 X +a0 be a polynomial with integer coefficients. Then the reduction mod p of f is the polynomial f = am X m + am−1 X m−1 + · · · + a1 X + a0 . 4 Proposition. Let f ∈ Z [X] of positive degree whose leading coefficient is not divisible by the prime p.

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Algebra 3: algorithms in algebra [Lecture notes] by Hans Sterk

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