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Problem in the style of M. Andronache

Source: 2010 contest

October 28, 2019
abstract algebraRing Theorypolynomialpolynomial ringbeautiful

Problem Statement

Let be a ring R R which has the property that there exist two distinct natural numbers s,t s,t such that for any element x x of R, R, the equation xs=xt x^s=x^t is true. Show that there exists a polynom in R[X] R[X] of degree ∣s−t∣(1+∣s−t∣) |s-t|\left( 1+|s-t| \right) such that all the elements of R R are roots of it.
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