TITLE: What's so special about characteristic 2?
QUESTION [52 upvotes]: I've often read about things which do not work in a field with a characteristic $2$, mainly things which have to do with factoring, or similar things. I'm not exactly sure why, but the only example of such a field I could think of is $\mathbb{Z}/2\mathbb{Z}$, which itself is an interesting field because it contains only the identity elements for the two groups, and naturally, it is a cyclic field. Do these properties lead to the fact that many things don't work if the charateristic is $2$
Any examples of things which break in such a field are also welcome.

REPLY [2 votes]: Normally, in a field, each element with a square root (other than zero) has two of them: x2-a2 = (x+a)(x-a), so both a and -a are roots. So by the pigeonhole principle, in a finite field (of odd characterisitic) half the nonzero elements have two square roots, and the other half have none. But in characteristic 2, a = -a, so all the elements have exactly one square root.
If you know the bare basics of elliptic curve cryptography, you might think this would make ECC impossible in fields of characteristic 2, but that isn't the case.
Some polynomial factoring algorithms over finite fields work more simply when the characterisitic is odd, just because 2 is a divisor of the number of nonzero elements (which is the order of the cyclic multiplicative group of the field). For characteristic 2, with size 22k, 3 is a divisor, and those factoring algorithms work with little modification. But with size 22k+1, 22k+1-1 might even be prime, requiring more major modifications for efficient factoring.