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Qiaochu_Yuan comments on Open Thread, April 15-30, 2013 - Less Wrong
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The theorems work out nicer if you don't. A field should be a ring with exactly two ideals (the zero ideal and the unit deal), and the zero ring has one ideal.
Ah, so it's for exactly the same reason that 1 isn't prime.
Yes, more or less. On nLab this phenomenon is called too simple to be simple.