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The structure of the multiplicative groups of Z/pZ or of Zp is the same for odd primes, but not for 2. Quadratic reciprocity has a uniform statement for odd primes, but an extra statement for 2. So in these examples characteristic 2 is a messy special case.

On the other hand, certain types of combinatorial questions can be reduced to linear algebra over F2, and this relationship doesn't seem to generalize to other finite fields. So in this example characteristic 2 is a nice special case.

Is anything deep going on here? (I have a vague idea here about additive inverses and Fourier analysis over Z/2Z, but I'll wait to see what other people say.)

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I wonder if you have any insight or conceptual explanation for why odd order groups are all solvable. – Gil Kalai Dec 5 '09 at 18:59
For the combinatorial point of view I guess that the importance of 2 is linked to the fact that for every set $X$ there's the canonical representation of each subset of $X$ with an element of $2^X$ which by the way is the support of a $F_2$ vector space, the space $\prod_{x \in X} \mathbb Z/2 \mathbb Z$. About other anomalies related to characteristic 2, I wonder if they are due to the fact that in characteristic 2 inverse coincide with identity and so we don't have the (involutive) symmetry respect to 0. – Giorgio Mossa Dec 10 '12 at 11:41

12 Answers 12

up vote 42 down vote accepted

I think there are two phenomena at work, and often one can separate behaviors based on whether they are "caused by" one or the other (or both). One phenomenon is the smallness of 2, i.e., the expression p-1 shows up when describing many characteristic p and p-adic structures, and the qualitative properties of these structures will change a lot depending on whether p-1 is one or greater than one. For example:

  • Adding a primitive p-th root of unity z to Qp yields a totally ramified field extension of degree p-1. The valuation of 1-z is 1/(p-1) times the valuation of p. This is a long way of saying that -1 lies in Q2.
  • The group of units in the prime field of a characteristic p field has order p-1. This is the difference between triviality and nontriviality.
  • As you mentioned, some combinatorial questions can be phrased in Boolean language and attacked with linear algebra.

The other phenomenon is the evenness of 2. Standard examples:

  • Negation has a nontrivial fixed point. This gives one way to explain why there are 4 square roots of 1 mod 2^n (for n large), but only 2 in the 2-adic limit. If you combine this with smallness, you find that negation does nothing, and this adds a lot of subtlety to the study of algebraic groups (or generally, vector spaces with forms).
  • The Hasse invariant is a weight p-1 modular form, and odd weight forms behave differently from even weight forms, especially in terms of lifting to characteristic zero, level 1. This is a bit related to David's mention of abelian varieties - I've heard that some Albanese "varieties" in characteristic 2 are non-reduced.
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Maybe this isn't very high concept, but I've always thought the "original sin" of 2 was that there's a integer which is a second root of unity, which doesn't happen for any other prime.

Why is this deep? Well, one way to think of it as this: in fields of characteristic p, pth roots of unity must all be trivial (and in general, taking pth roots is a bad idea), so fields of characteristic 2 are particularly incompatible with the integers, since they have to destroy -1.

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Yes, that's more or less what I was trying to say in my parenthetical comment. But I can't decide whether this is deep or whether it's just because historically mathematicians happen to like additive inverses. – Qiaochu Yuan Oct 17 '09 at 20:57
I wouldn't think that mathematicians study additive inverses because we like them. Rather it's because that's what nature often gives us. – Kevin H. Lin Oct 18 '09 at 1:31

I think 2 is not special, we just see the weirdness at 2 earlier than the weirdness at odd primes.

For example, consider ExtE(x)(Fp , Fp) where E(x) denotes an exterior algebra over Fp. If p=2 this is a polynomial algebra on a class x1 in degree 1 and if p is odd this is an exterior algebra on a class x1 tensor a polynomial algebra on x2. I say these are the same, generated by x1 and x2 in both cases and with a p-fold Massey product < x1,...,x1>=x2. The only difference is that a 2-fold Massey product is simply a product.

In what sense are the p-adic integers Zp the same? One way to say it is that if you study the algebraic K-theory of Zp you find that the first torsion is in degree 2p-3. If p=2 this is degree 1, and K1(A) measures the units of A (for a reasonable ring A). If p is odd it measures something something more complicated. Another way to say it is that Zp is the first Morava stabilizer algebra and there is something special about the n'th Morava stabilizer algebra at p if p-1 divides n. If you study something like topological modular forms, this means the primes 2 and 3 are special.

The dual Steenrod algebra is generated by \xii at p=2 and by \xii and \taui at odd primes. But really it is generated by \taui with a p-fold Massey product < \taui,...,\taui>=\xii+1 at all primes, after renaming the generators at p=2. (Again a 2-fold Massey product is just a product.)

I could go on, but maybe this is enough for now.

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I like this answer. Thanks! – Qiaochu Yuan Oct 23 '09 at 16:21
This is a completely subjective comment, but: I would say that the fact that $2$ starts acting weird much earlier in the day than the other primes is a special property of $2$. For instance, the theory of quadratic forms has a radically different feel in characteristic $2$. By analogy one would look to, say, cubic forms in characteristic $3$, but (i) the theory of cubic forms is not nearly as well-developed as that of quadratic forms, and (ii) in the best understood case -- elliptic curves -- characteristic $3$ is a little strange but characteristic $2$ is still stranger! – Pete L. Clark Jun 30 '10 at 7:42
@Vigleik: This is true in so much of homotopy theory! For example, $p \cdot id_{S/p}$ for $S/p$ the mod-$p$-Moore spectrum is non-trivial exactly for $p=2$. This has been generalized by Stefan Schwede to the fact that $S/p$ has order at most $p-2$ in a certain sense and order $\geq 1$ means that $p\cdot id = 0$. See Theorem 1 of… – Lennart Meier Aug 28 '13 at 17:58

$x\mapsto x^2$ is a (1-1) automorphism on fields of characteristic 2, whereas it is 2-1 on $F_q\setminus{0}$ if q is odd. Not a high level concept, but this is where all things quadratic (reciprocity, residues, etc.) break down.

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It's anthropocentrism. If we were starfish, we would think that the prime $5$ was the weird one.

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I was about to post a similarly spirited answer, adorned with the observation that I think that 2 looks as special as it does because we have so far concentrated mainly in studying situations where it is special (quadratic stuff), not because they are the only ones of interest but because they are the ones we can deal with. Other primes do already show up in very peculiar ways when dealing with certain types of objects (an example is Lie theory, where 3 is also special) and I would not be surprised if this would occur more often in the future in other areas. IOW, starfish are smarter than us. – Mariano Suárez-Alvarez Jun 30 '10 at 11:21
Certainly if starfish had a god, he'd have five sides (misquoting Montesquieue). And certainly when they need a tiling of the seabed, they will soon proclaim that 5 is indeed a weird prime, and some will state that the mentioned god chose it to not allow them to fill the floor. But then there are some of them who meditate in their spare time and realise that if their symmetries were more of order 2 or 3 -- like in some of the primitive species they observe -- they would have better chances to do so. And some of them will play with the thought that ... – Torsten Schoeneberg Mar 21 '14 at 19:49
(cont'd) indeed 3, 4 and 6 are exceptional numbers when it comes to matters of seabed-tiling, and the speciality of 5 is just that it's the first of the non-weird numbers (except for 2, but hey, what straight starfish cares about 2?). - Then some will say that exactly this shows the weirdness of 5 (if not the supremacy of starfishdom), but others will reply that, you know the paradox, there is no smallest non-interesting number, and every number looks weird when you look at it long enough. But still, it makes a curious starfish wonder, some things calmly tile the sea floor and others don't ... – Torsten Schoeneberg Mar 21 '14 at 19:51

Most of the examples of char 2 being funny that I know of (and there are a lot) really boil down to the fact that x = -x.

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This is pretty much what I meant, actually. – Harrison Brown Oct 17 '09 at 20:47
That said, this feels to some degree like begging the question, since there's no obvious a priori reason (to my mind) why x = -x should be special, while x + x = -x, x + x + x + x = -x, etc., should all act essentially the same. Certainly there's a difference, but it feels more like one of degree than of kind. – Harrison Brown Oct 17 '09 at 20:54
Well, involutions are more common in mathematics than other kinds of automorphisms, and so the equation x = -x just appears more often. (For instance abelian varieties always have the map [-1]) – David Zureick-Brown Oct 17 '09 at 21:15
Funny that the first examples that come to your mind are abelian varieties! How about abelian groups!? :) – Kevin H. Lin Oct 18 '09 at 0:48
Both abelian varieties and abelian groups also have the map [n]. – jmc May 18 '13 at 19:50

Here is my computational reason (instead of a high-concept explanation) why the primes 2 and 3 are special (hypotheses of many theorems on algebraic groups, linear or projective exclude both these primes).

The numbers 2 and 3 got into the Primes Club by 'dubious' means!

Given $p$, to certify it as a prime a number, we need to check no number $d $ with $1 < d \leq [ \sqrt p ]$ divides it. For 2 and 3 this condition is vacuously true as there are no integers in that interval, wheres 5 onwards they really needed to pass the test!

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I think one reason that makes 2 special is that, for the only archimedean place of the rationals, namely the real numbers, it has absolute Galois group of order 2.

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In other contexts, maybe, but I don't see what this has to do with the properties of, say, Z/pZ. – Qiaochu Yuan Oct 18 '09 at 0:40
I think that is a statement about real closed fields rather than characteristic two phenomena (although it is an interesting observation involving the number 2). [Note: Real closed fields can be characterized by several equivalent properties. One of them is that the algebraic closure is a nontrivial finite degree extension. Another is that a real closed field admits an ordered field structure and any odd-degree polynomial has a root.] – S. Carnahan Oct 18 '09 at 2:32
I think that this indeed related to the fact that Z/2Z stands out of other Z/pZ's (hopefully not in a very superficial sense). For example, the natural question about orientability of a real fibre bundle lead to consideration of cohomology groups with coefficients in Z/2Z (Stiefel–Whitney classes). – Pasha Zusmanovich Nov 1 '09 at 12:01

When I was a lad, I was taught that 2 acted strange (compared to other prime numbers) because it was of the form $1-u$ for a unit, $u$. I guess the test of whether this holds water would be something like: let $R$ be a (commutative) ring (with unity), then do the irreducibles in $R$ of the form $1-u$, $u$ a unit in $R$, stand out from the other irreducibles in $R$ in some significant way?

I think there is some evidence in favor of this hypothesis in the rings ${\bf Z}[\rho]$, where $\rho$ is a $p$th root of unity, and $1-\rho$ is an irreducible that may require special consideraiton.

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This is certainly not very high-concept, but I've always just thought there was more "room to maneuver" in high characteristic, and there's some inexplicable (well, nearly inexplicable!) "phase change" that happens between characteristic 2 and 3. And yeah, this is fuzzy and not well-defined, but it's nevertheless how I think about it?

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A more succinct way to put this is that Z/2Z doesn't have any non-identity elements; in other words, 2 is "degenerate." But for some reason I don't find this very satisfying. – Qiaochu Yuan Oct 17 '09 at 20:59

I think the simplest answer is just that... F_2 is much simpler to analyze and use, and for example has the concrete interpretation of also being a nice way to have permutation matrices defined over a field, as well as being a cute way of encoding statements in boolean logic.

In short, I don't think any of this is emblematic something deep, but rather that F_2 is the easiest to understand of all the finite fields, and its simplicity makes it very useful in works that call for a finite field

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Hmmm, I don't think you and the author interpreted the word "special" in very similar ways. – Ben Webster Oct 17 '09 at 20:35

I see binary arithmetic to be the natural companion to set theory, and this singles F_2 out given that set theory is the core of mathematics. The basic idea being that all subsets of a finite set with n elements can be associated with a binary n-tuple (an element of (Z_2)^n). Or vice-versa, since we could as well consider set theory as the study of binary n-tuples. (Just an elementary example: the sum of two such n-tuples, using 1+1=0, corresponds to the symmetric difference of the two sets). The very fact that a set of n elements has 2^n subsets reminds us of the core meaning of the powers of 2.

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Right, but as far as I can tell it's a "coincidence" (whatever that means) that arithmetic over F_2 happens to describe XOR and AND. There just aren't that many commutative and associative operations {0,1}^2 -&gt; {0,1}. – Qiaochu Yuan Oct 23 '09 at 14:20
Topos theory (/constructive set theory), corroborates the idea that this is a coincidence: in that setting, there are more truth values than just true and false (the ``object of truth values'' is the subobject classifier \Omega = \powerset(1)) but F_2 remains as just {[0],[1]}, so in that setting, \Omega \notiso F_2. – Peter LeFanu Lumsdaine Jun 4 '10 at 9:01

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