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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 (https://en.wikipedia.org/wiki/Kronecker%E2%80%93Weber_theorem). 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?

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    $\begingroup$ It also has an odd totient (1) and the primitive root is the identity. All sorts of small-number issues. $\endgroup$ – Joffan Jul 3 '18 at 16:53
  • $\begingroup$ 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. $\endgroup$ – Hagen von Eitzen Jul 3 '18 at 16:58
  • $\begingroup$ 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 $2$. $\endgroup$ – Elliot G Jul 3 '18 at 17:01
  • $\begingroup$ @Hagen von Eitzen: Are you sure your first identity allways holds? $\endgroup$ – PaulTaylors Jul 3 '18 at 17:01
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    $\begingroup$ I asked a similar question on MO long ago: mathoverflow.net/questions/915/… $\endgroup$ – Qiaochu Yuan Jul 3 '18 at 19:31

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