I have proved that 1 + 1 = 0 [closed]

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I have proved that 1 + 1 = 0 in one of my questions where a field was given. I was wondering if it is true in every field we have 1 + 1 = 0. Also i was wondering (i know how to prove 1 + 1 = 0) can you prove 0 + 0 = 1 the same way as 1 + 1 = 0?

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1 Answer

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It is certainly not the case that 1+1=0 in every field; e.g., the rationals, the reals, the complex numbers, the $p$-adics, the integers modulo 17, . . .

There are fields in which 1+1=0 - this property is called characteristic 2. More generally, for any prime $p$, there are fields in which 1+1+ . . . +1 ($p$ times) equals 0; see (algebra).

On the other hand, if 0+0=1, then we would have 0=1, which is prohibited by the field axioms.

On the other other hand, there are algebraic structures - non-unital rings (rngs) - in which 0=1 is possible; on the other other other hand, that only happens for the trivial rng, so it's not very interesting.

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