By Hans Sterk
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Additional info for Algebra 3: algorithms in algebra [Lecture notes]
2x = x2 + 1 1 1 + = log(x + i) + log(x − i) = log(x2 + 1). x+i x−i In this case no constants outside the rationals are necessary. But we still need a logarithm. 6 (Denominator is a pure power) Here we are dealing with the case a/bm with • deg(a) < deg(b), • b is squarefree, • m > 1. The idea in this case is to apply integration by parts to reduce the exponent in a/bm as far as possible, in fact to 1. Since b is squarefree, gcd(b, D(b)) = 1, and the Euclidean algorithm produces an identity of the form ub + vD(b) = a.
The Hensel lift aims at lifting a factorisation modulo pk to a factorisation modulo pk+1 . Together with the Landau– Mignotte bound on the coefficients of possible factors, these ingredients can be put together into an algorithm. 4 Theorem. (Hensel lift) Let p be a prime and let f, g, h ∈ Z [X] be monic polynomials of positive degree such that f ≡ g h (mod pm ) for some m ∈ N with gcd(g (mod p), h (mod p)) = 1. Then there exist monic polynomials ˜ such that g˜ ≡ g (mod pm ), h ˜ ≡ h (mod pm ) and f ≡ g˜ h ˜ (mod pm+1 ).
Let k be a field and let a ∈ k. If f ∈ k[X], then f is irreducible if and only if f (X +a) is irreducible. 8 Example. For p prime, define Φp (X) = X p−1 + X p−2 + · · · + X + 1. Then Φp (X) = Replace X by X + 1 and we find Φp (X + 1) = Xp − 1 . 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.