Re: None of this nursery school stuff - a proper maths problem. 25$ re
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Is there unique prime factorization in Z[sqrt(-17)]?
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Theorem 8.22 of An Introduction to Number Theory, Harold M. Stark:
If d<0, then Q(\sqrt{d}) has the unique factorization property if and only if d is one of the nine numbers -1,-2,-3,-7,-11,-19,-43,-67, and -163.
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Exactly. Unfortunately, -17 not equal to 1 mod 4, which effectively destroys any chance of the ring of integers of Q(sqrt(-17)) being a UFD.
(Briefly, this is because the ring is too small. Note that if a = c + d.sqrt(-17) and b = c - d.sqrt(-17) then (X - a)(X - b) = X^2 - 2cX + c^2 - 17d^2 as the minimum polynomial doesnt have many roots as T =( 1 + sqrt(-17) / 2 ) doesnt solve the above). So finding a euclidean norm is impossible here. For positive values in the sqrt matters get even worse...)
But its still possible to navigate around this. But as I said this problem is way too hard, which is my fault.
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