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I was trying to prove that if d^2 divides z^2, then d divides z.
It seems like it should be an easy proof, but the proof I came up with is long and unclear (maybe even wrong). Can anyone come up with a better proof?
Here it is:
By definition, means there is a positive integer n such that . Suppose n is not the square of an integer. Let , where m>1 has no square divisors. Let p be a prime divisor of m. We have , and since the powers of p in , , and are even, the power of p in m must be even. But then m has a square factor. By contradiction, therefore, n must be the square of an integer. Let . Then and since z and d are positive and you can choose a to be positive, z=ad, which is the definition of .