x being reducible in R[x]

Posted: December 1, 2009 in Elementary Algebra; Problems & Solutions, Rings and Modules
Tags: ,

There are many commutative rings R satisfying this property that the indeterminate x is reducible in the polynomial ring R[x]. Here are two examples: in (\mathbb{Z}/6\mathbb{Z})[x] we have x=(3x+4)(4x+3) and in (\mathbb{Z}/10\mathbb{Z})[x] we have x=(5x+4)(4x+5). In general, if a commutative ring R  has an idempotent e \neq 0,1, then

x=(ex + 1-e)((1-e)x + e).

So if an integer n > 1 has at least two distinct prime divisors, then x will be reducible in (\mathbb{Z}/n\mathbb{Z})[x].

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