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yet another condition for a ring to be a field

Source: Romanian Nationals RMO 2005 - grade 12, problem 4

March 31, 2005
inductiontopologyRing Theorysuperior algebrasuperior algebra unsolved

Problem Statement

Let AA be a ring with 2n+12^n+1 elements, where nn is a positive integer and let M={k∈Z∣k≥2, xk=x, ∀ x∈A}. M = \{ k \in\mathbb{Z} \mid k \geq 2, \ x^k =x , \ \forall \ x\in A \} . Prove that the following statements are equivalent: a) AA is a field; b) MM is not empty and the smallest element in MM is 2n+12^n+1. Marian Andronache
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