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number of perfect squares in a ring may make it a field

Source: RMO District 2005, 12th Grade, Problem 4

March 5, 2005
group theoryabstract algebracalculusintegrationalgebrafunctiondomain

Problem Statement

Let (A,+,⋅)(A,+,\cdot) be a finite unit ring, with n≥3n\geq 3 elements in which there exist exactly n+12\dfrac {n+1}2 perfect squares (e.g. a number b∈Ab\in A is called a perfect square if and only if there exists an a∈Aa\in A such that b=a2b=a^2). Prove that a) 1+11+1 is invertible; b) (A,+,⋅)(A,+,\cdot) is a field. Proposed by Marian Andronache
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