2014-08-29 10:51 UTC+02:00, John Cremona <[email protected]>: > 1 is definitely not a prime power. It's basically the same reason that 1 > is not a prime. > > Some reasons: (1) positive integers are uniqely products of prime powers > (with 1 being the empty product!) Uniqueness would fail if 1 were allowed. > (2) a positive integer n is a prime power iff nZ is a primary ideal of Z > (that got you 2 marks in an exam I set this year), and the definition of > primary ideal (as with prime ideal) explicitly requires the ideal to be > proper. (3) q is a prime power iff there exists a field of cardinality q
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