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By Hans Sterk

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X The right–hand side works out as X p−1 + p p X + p. X p−2 + · · · + 2 p−1 This polynomial is suitable for the application of Eisenstein’s criterion for the prime p. We conclude that Φp (X) is irreducible. Φp (X) is part of a family of polynomials, the cyclotomic polynomials Φm (X) for m ∈ Z, m > 0. These are the minimal polynomials of e 2πi m and of fundamental importance in number theory. 1 Factoring a polynomial in Fq [X] (with q a power of the prime p) is a finite job. The purpose of this section is to demonstrate Berlekamp’s algorithm, a more efficient way of factoring.

If this is not the case, then the original integral is not elementary; otherwise, we proceed to the equations with j = 0. These are of the form y + f y = g, with f, g ∈ K. This differential equation is called the Risch differential equation. We need to solve it within K. If any of these differential equations fails to have a solution in K, then p(θ) is not elementary. Solving the Risch equations brings us into the realm of symbolic differential equations, the topic of one of the projects for this course.

I From this equality we deduce that b|v1 · · · vn . The converse also holds: since vj |ui for j = i, the same equality implies that vj |bvj uj . But gcd(vj , vj ) = 1 since vj is squarefree and gcd(vj , uj ) = 1 by construction, and so vj |b. Again using that the vi are relatively prime, we conclude that the product v1 · · · vn divides b. Using the equality b = v1 · · · vn , we find a = i ci vi ui . Then a − cj b = i c i v i ui − c j v i ui = i i (ci − cj )vi ui . ), so that vj is a common divisor of a − cj b and b.

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


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