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eisenstein-irred

# eisenstein-irred - Eisenstein Criterion for Irreducibility...

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Eisenstein Criterion for Irreducibility of a Polynomial : Algebra Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] 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, α = A 0 + A 1 x + A 2 x 2 + · · · + A J x J , β = B 0 + B 1 x + · · · + B K x K , μ = M 0 + 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 α 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 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 | n means “ k does not divide n .” Use n |• k and n | 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 μ is irreducible over Z if, whenever it can be factored into intpolys μ = αβ , then either α or β 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 μ is good if its constant term is non-zero. The reversal of a deg- L good μ is M L + M L - 1 x + M L - 2 x 2 + · · · + M 1 x L - 1 + M 0 x L , denoted by - μ . On the set of good polys, “reversal” is an involution.

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eisenstein-irred - Eisenstein Criterion for Irreducibility...

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