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Unformatted text preview: Eisenstein Criterion for Irreducibility of a Polynomial : Algebra Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA firstname.lastname@example.org 14 February, 2008 (at 09:47 ) Abstract: Proofs of Eisenstein Criterion and the Gauss Lemma. Nomenclature. Use * poly for polynomial and coeff for coefficient. I will be considering three polys, = A + A 1 x + A 2 x 2 + + A J x J , = B + B 1 x + + B K x K , = M + M 1 x + M 2 x 2 + + M L x L , with each of A J ,B K ,M L non-zero. 1 : My convention is that later coefficients are all defined, and equal zero; so 0 = M L +1 = M L +2 ,... . An int- poly is a poly whose coefficients are integers. A rat- poly has rational coefficients. Traditionally, the sym- bol Z [ x ] is used for the set of intpolys, and Q [ x ] for the collection of ratpolys. These sets are rings. A poly is a unit in Z [ x ], if its reciprocal is also in Z [ x ]. There are only two units in Z [ x ]; the constant polys 1. In Q [ x ], however, each constant poly x 7 q is a unit, where q ranges over the non-zero * Use N to mean congruent mod N . Let n k mean that n and k are co-prime. Use k | n for k divides n . Its negation k r | n means k does not divide n . Use n | k and n r | k for n is/is-not a multiple of k . Finally, for p a prime and E a natnum: Use double-verticals, p E || n , to mean that E is the highest power of p which divides n . Or write n || p E to emphasize that this is an assertion about n . Use PoT for Power of Two and PoP for Power of ( a ) Prime . rationals. A poly is irreducible over Z if, whenever it can be factored into intpolys = , then either or is a unit ( in Z [ x ] ). Thus 6 x- 15 is reducible over Z , since 6 x- 15 = 3 [2 x- 5] . 2 : However 6 x- 15 is ir reducible over Q , since 3 is a Q-unit. Reversing a poly. Below is a trivial factoring re- sult; I put it here so that we can apply the Eisenstein criterion to it later. Say that a poly is good if its constant term is non-zero. The reversal of a deg- L good is M L + M L- 1 x + M L- 2 x 2 + + M 1 x L- 1 + M x L , denoted by - . On the set of good polys, reversal is an involution....
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