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A Chebotarev-free argument is given by our own @Pete L Clark here:
http://math.stackexchange.com/questions/6976/proving-that-an-integer-is-the-n-th-powerhttps://math.stackexchange.com/questions/6976/proving-that-an-integer-is-the-n-th-power
http://math.stackexchange.com/questions/6976/proving-that-an-integer-is-the-n-th-power
https://math.stackexchange.com/questions/6976/proving-that-an-integer-is-the-n-th-power
