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

By Hans Sterk

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The integrand is θ/x, with θ = log(x). Set θ = q2 θ 2 + q1 θ + q 0 . x Then we get the system q2 = 0, (2q2 θ + q1 ) = 1/x, q1 θ + q0 = 0. ) From the second equation we conclude that 2q2 θ + q1 = θ+γ (where γ is a constant), and therefore q2 = 1/2 and q1 = γ. Substituting this in the third equation yields q0 = −γθ + δ, where δ is a constant, so that γ = 0 since q0 is does not depend on θ. So q0 is a constant and we get log(x) 1 1 = θ2 + constant = log(x)2 + constant. 1 Definition. A group G is a nonempty set with a binary operation that satisfies: • associativity: g(hk) = (gh)k for all g, h, k ∈ G.

U We often write θ = log(u). The extension L : K is said to contain an exponential of u ∈ if it contains an element θ such that D(θ) = D(u) · θ. We often write eu or exp(u) for such θ. 10 Note that we haven’t proved that given u ∈ K an extension containing a logarithm or exponential exists. Also, we are not claiming that such logarithms or exponentials necessarily bring us outside the field K. The notation log(u) and eu suggests that such expressions share more of the properties of the logarithm and exponential, respectively.

X −1 (X + 1)p − 1 . 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.

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