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But Z[sqrt(-3)] is not the ring of Eisenstein integers.

Regardless, the fact that Z[sqrt(-n)] is a UFD for n = 1,2 but not for n >= 3 is indeed quite counter-intuitive. But Gowers isn't asking whether it's intuitive for Z[sqrt(-n)]. He's asking about the positive integers, and the positive integers have a very special well-ordering which respects the arithmetic operations and for which humans have quite a powerful intuition.

I maintain that Z[sqrt(-n)] is more different from Z than Gowers is letting on and that if you actually look at all the differences, rather doing the opposite and trying to make it seem similar to Z, you'll quickly realize why people feel that the FTA is not very bizarre for Z but it is for Z[sqrt(-n)].




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