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I have been asking to myself for a while now why $2$ has such an exceptional behaviour in algebraic number theory. For example, the Kronecker-Weber Theorem proof was completed for all cases but that of number fields of degree a power of 2 by Kronecker, and a full proof was found by Neumann/Hilbert only after $40/50$ years. Often it is the splitting behaviour of $2$ that is exceptional and troublesome, for example it is the only prime $p$ such that there are more than one quadratic field of conductor a power of $p$ (they are $\mathbb{Q}(i),$ of conductor $4,$ and $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{-2})$, of conductor $8$). Maybe the most famous example is that $x^p+y^p=z^p$ has non trivial integer solutions only for $p=2$. I think there are many more examples in number theory of this exceptional behaviour. Everything I could think of as a "philosophical" explanation is the trivial observation that $2$ is the only even prime (which I find a bit circular) and that in any case it is so just because.

So, my question is: intuitively, why is $2$ so troublesome?

Miss Mae
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PaulTaylors
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    It also has an odd totient (1) and the primitive root is the identity. All sorts of small-number issues. – Joffan Jul 03 '18 at 16:53
  • In general, $(np+1)^2\equiv 1\pmod {p^2}$ only and not better, but always $(2n+1)^2\equiv 1\pmod{2^3}$. Of course this is due to squaring (instead of cubing etc.), but squaring is so much more relevant than other powers, for example because it happens every time we plug in identical inputs into a bilinear form. – Hagen von Eitzen Jul 03 '18 at 16:58
  • The number $2$ appears a lot, perhaps second only to the number $1$, so it is hard to do algebra in characteristic $0$ because you can never divide by $2$. – pancini Jul 03 '18 at 17:01
  • @Hagen von Eitzen: Are you sure your first identity allways holds? – PaulTaylors Jul 03 '18 at 17:01
  • There is also the fact that modulo 2 all numbers are square, i.e. 2 is exceptional even for the quadratic reciprocity law. One could object that this is a bit circular as well, because we are considering the reciprocity law for 2nd powers. – PaulTaylors Jul 03 '18 at 17:06
  • Out of all the properties mentioned, I cannot but observe that the only properties which do not explicitly or implicitly mention 2 and characterize 2 among the primes is the second one mentioned by Joffan and the last fermat theorem for prime exponents. Another such property is that 2 is the smallest prime, which is not very arithmetical in nature, though. – PaulTaylors Jul 03 '18 at 17:18
  • See also https://math.stackexchange.com/a/1177352/589 – lhf Jul 03 '18 at 17:35
  • Fields of characteristic 2 are odd places because everything is its own additive inverse, so you lose out on the standard proof of showing x=-x implying x=0. – Alan Jul 03 '18 at 17:41
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    I asked a similar question on MO long ago: https://mathoverflow.net/questions/915/is-there-a-high-concept-explanation-for-why-characteristic-2-is-special – Qiaochu Yuan Jul 03 '18 at 19:31
  • Also related is this MO question: https://mathoverflow.net/q/160811/27465 – Torsten Schoeneberg Dec 01 '20 at 03:27

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