When we start looking at fields, there are a collection

of properties that are interesting. The simplest one – and

the one which explains the property of the nimbers that

makes them so strange – is called the

*characteristic* of the field. (In fact, the

characteristic isn’t just defined for fields – it’s defined

for rings as well.)

Given a field F, where 0_{F} is the additive

identity, and 1_{F} is the multiplicative identity,

the characteristic of the field is 0 if and only if no

sequence of adding 1_{F} to itself will ever result

in 0_{F}; otherwise, the characteristic is the

number of 1_{F}s you need to add together to get

0_{F}.

That sounds confusing – but it really isn’t. It’s just

hard to write in natural language. A couple of examples will

make it clear.

The field of nimbers has characteristic two – because

1+1=0 in the nimbers – so adding to 1s together gives you

0.

The field of numbers modulo-7 is a nice simple field,

with characteristic 7: in mod-7, you need to add 1+1=2;

2+1=3; 3+1=4; 4+1=5; 5+1=6; 6+1=0. So 1+1+1+1+1+1+1=0.

The real numbers have characteristic 0, because adding

one repeatedly will never give you 0. In fact, the

characteristic is 0 for any *ordered* field O – that

is, a field with a properly defined “<” operator (one

where ∀x∈O, ∃y∈O : x<y ∧

∃z∈0 : z<y, and < is total, transitive,

antisymmetric, and antireflexive).

See? Not hard, right?

What makes nimbers so strange is that they’re an

infinite set with characteristic two. Our intuitions about

numbers all rely on the structure of an infinite set with

characteristic 0. All of the strange properties of nimbers

can be traced back to the fact that their basic structure is

strange; and the root of that strangeness always traces back

to the field characteristic.

There’s an interesting notion also associated with the

characteristic of a field. Like groups could have

sub-groups, fields can have sub-fields: a sub-field of a

field (F,+,×) is a field (G,+,×) where G⊆F.

For every field, there is a minimal subfield – a field where

removing any elements would require removing 1 to maintain

closure. The minimal subfield of a field, F, is called the

*prime subfield* of F.

Here’s where we get something fascinating. There are

numerous limits to the kinds of structures that can fulfill

the field axioms to become proper fields. One of the amazing

ones (at least to me) is that every field has either

characteristic 0 (in which case it’s isomorphic to a

sub-field of the complex numbers), or it has a

*prime* characteristic, and it’s prime subfield is

isomorphic to a finite field whose size equals it’s

characteristic.

What does this tell us? Well, looking at it

philosophically, it means that the prime numbers are very

deeply embedded in the structure of algebra. Even if we

throw away our standard number system and go into the realm

of the abstract, the numbers that are prime in the basic set

of natural numbers remain fundamental. Both the concept and

the specific values of the prime numbers are deeply embedded

all the way down in the foundations of what we know as

mathematics. Even when we start from scratch, with abstract

sets, and build upwards into mathematical structures, when

we get to the point where we can build things that behave

even just a little bit like numbers, the prime numbers are

inevitable – even though they aren’t necessarily prime in

every field!

After all, think about the nimbers! The number 11 for example, is clearly prime in the natural numbers. But in nimbers, it’s 8×4: and yet, even if we play with the nimbers, we’ll wind up finding out that 11 is prime.

Incidentally, I’m still trying to hack together an implementation of the nimbers. As one of the commenters (Xanthir?) mentioned, it’s hard to get them right, and I haven’t had much free time to try it. I’ve got a bunch of Scheme code that’s close, but it’s still got one recursive case that never terminates. Nimbers are truly annoying.

Coinevery field has either characteristic 0 (in which case it’s isomorphic to a sub-field of the complex numbers)Hmm…

I’m trying to figure out how to reconcile this statement with the idea of a field whose set cardinality is greater than that of the reals– such a field clearly couldn’t be isomorphic to a subfield of the complexes.

Is the idea that no such field exists? Or that if such a field existed its cardinality could not be zero?

Mark C. Chu-CarrollCoin:

A field is a *set* of values. If something has cardinality greater than the reals, it’s not a set – it’s a proper class.

It’s a bit of a weasel in some ways, but a lot of field theory won’t work for proper classes – in the basic logic of the proofs, it relies on ZFC.

AntendrenIf something has cardinality greater than the reals, it’s not a set – it’s a proper class.This is wrong. Recall Cantor’s proof that P(X) has cardinality strictly greater than that of X. Further, ZFC says that if X is a set, then P(X) is a set. R (the reals) is a set. So P(R) is a set with cardinality strictly greater than that of R.

There are fields of all infinite cardinalities. The correct statement about fields being subfields of C is “every field of characteristic 0 and cardinality no greater than that of the reals is isomorphic to a sub-field of the complex numbers.”

Joshua ZuckerIf you’re trying to implement Nimbers in scheme, you might find

http://home.att.net/~numericana/answer/fields.htm

useful — search for the mention of nimbers — just below the multiplication table, there’s a nice suggestion of how to implement the multiplication recursively to run in fairly reasonable time.

Joshua ZelinskyAntendren is correct; there are fields which are of small cardinality that are characteristic zero and are not isomorphic to a subset of C. Consider for example the ring S) that is formed by C adjoint |P(R)| distinct formal variables (where P(R) is the powerset of the real numbers). Take the Field of fractions of S (we can do this because S is Dedekind). This resulting field is small enough that we can represent it as a set but it has cardinality |P(R)| which is greater than the cardinality of C, so this field cannot be isomorphic to a subfield of C.

AnonymousIf something has cardinality greater than the reals, it’s not a set – it’s a proper class.It’s a bit of a weasel in some ways, but a lot of field theory won’t work for proper classes – in the basic logic of the proofs, it relies on ZFC.Not only is the first statement wrong (as Antendren pointed out), but the second is as well. I wouldn’t swear that nothing ever breaks down, but in general field theory works fine for proper classes. This is usually an irrelevant fact, since it’s very rare to find a “naturally occurring” example of a field that’s a proper class. The only example I’ve ever run across is Conway’s field of surreal numbers, and even there you can do a version using sets if you’d like (that’s just not how Conway did it, because for example he wanted to include all ordinals).

Strictly speaking, the word “field” doesn’t apply to proper classes, since it is almost universally defined in terms of sets, but I don’t know of a compelling reason why it couldn’t.

EricWouldn’t the numbers modulo-6 make just as good a field as numbers modulo-7, with characteristic 6 (ie neither 0 nor prime)?

JCI would like if it were possible a proof, proving that

if finite field does not have characteristic zero than i must have charateristic prime.

AnonymousWouldn’t the numbers modulo-6 make just as good a field as numbers modulo-7, with characteristic 6 (ie neither 0 nor prime)?Modulo 6, the numbers 2 and 3 are not zero but don’t have multiplicative inverses, so you don’t get a field.

I would like if it were possible a proof, proving thatif finite field does not have characteristic zero than it must have charateristic prime.

In any field, when the product of two numbers is zero, one of the numbers must be zero itself. Now suppose you have characteristic n. In other words, the sum 1+1+…+1 (with n 1’s) vanishes. If n factors, then this sum factors as well. For example, because 2*3=6, (1+1)*(1+1+1)=1+1+1+1+1+1. If n factors, then one of these factors must vanish. However, by definition the characteristic is the smallest possible number of 1’s that sum up to zero. That means it must be prime, since any nontrivial factor would be even smaller.

Notice that this breaks down modulo 6. Mod 6, we have 2*3=0, but neither 2 nor 3 vanishes.

Darin BrownMark, I find it amazing that you continue to spew patent nonsense, confusing explanations, and constipated thought week after week, month after month, year after year, like

“For every field, there is a minimal subfield – a field where removing any elements would require removing 1 to maintain closure.”

COMMENT: Why don’t you just say “it’s the smallest subfield containing 1”?“What makes nimbers so strange is that they’re an infinite set with characteristic two.”

COMMENT: The nimbers are strange for many reasons, but that isn’t really one of them. There aremuchless exotic fields, e.g. (Z/2Z)(x), the rational function field overZ/2Z, which are infinite and have characteristic 2.“Even when we start from scratch, with abstract sets, and build upwards into mathematical structures, when we get to the point where we can build things that behave even just a little bit like numbers, the prime numbers are inevitable – even though they aren’t necessarily prime in every field!… After all, think about the nimbers! The number 11 for example, is clearly prime in the natural numbers. But in nimbers, it’s 8×4: and yet, even if we play with the nimbers, we’ll wind up finding out that 11 is prime.”

COMMENT: I have no idea what this is supposed to mean. Fields don’t have prime elements, a prime element is a non-zero non-unit, and every non-zero element of a field is a unit.“In fact, the characteristic is 0 for any ordered field O – that is, a field with a properly defined ‘LT’ operator [“less than”, couldn’t get the symbol to process] (one where ∀x∈O, ∃y∈O : x LT y ∧ ∃z∈0 : z LT y, and LT is total, transitive, antisymmetric, and antireflexive).”

COMMENT: I can’t even make sense of this so-called “definition”. Assuming your “O”‘s [letter O] and “0”‘s [numeral 0] are the same, first of all, “LT” is not an “operator”, it’s called a “relation” (you seem to be constantly confused about even the most elementary mathematical terms). Also, I hate to break it to ya, but every total relation is trivially necessarily reflexive, not “antireflexive”, whatever the hell that means (I think you mean “irreflexive”??) And what the hell does “∀x∈O, ∃y∈O : x LT y ∧ ∃z∈0 : z LT y” mean?? “For all x in O, there is a y in O such that (x LT y and there is a z in O such that z LT y)”?? Doesn’t the “x” in “x LT y” itself satisfy “there is a z in O such that z LT y”? And is anyone else bothered by the fact that Mark’s “definition” of an ordered field makes no reference to axioms combining the order and field operations??“every field has either characteristic 0 (in which case it’s isomorphic to a sub-field of the complex numbers)”

COMMENT: Simply false. I think you’re missing a hypothesis somewhere.“If something has cardinality greater than the reals, it’s not a set – it’s a proper class.”

COMMENT: I can understand someone goofing on the previous statement, but to make this statement demonstrates profound mathematical illiteracy.“Arrow composition has to meet two properties, which are going to look pretty familiar from group theory: associativity, and identity. Associativity says that aº(bºc)=(aºb)ºc; and identity says that that for any arrow f:a→b, there are arrows 1a and 1b such that 1bºf = f = fº1a.”

COMMENT: I think you really mean to say that for every object a, there is an identity arrow 1a: a→a such that for every arrow f:a→b, 1bºf = f = fº1a. But as usual, you got a bit tongue-tied.“For an example that’s important for looking at group theory related topics, we say than an arrow is iso (i.e., it is a iso-morphism) if it’s reversable – which in category theoretic terms means that if f is an iso-arrow, then there is an arrow f-1 such that fºf-1=1.”

COMMENT: Again, I think you mean, f:a→b is iso if there is f-1:b→a such that f-1ºf=1a and fºf-1=1b. You must have been tongue-tied again.“That’s a common error – I can’t count how many CS papers I’ve seen that show a diagram without proving that it actually commutes!”

COMMENT: Or maybe you just don’t understand what you’re reading. Not every diagram commutes. That’s why we call the ones that DO commutecommutative diagrams. Go figure.“For an example of a diagram, there’s a type of arrow between two objects A and B called a principle morphism. The idea of a principle morphism is that a morphism M from A to B is principle if and only if every self-arrow of A (or endomorphism, an arrow from A to A), composed with an arrow from A to B yields M. As a very visual thinker, I find that paragraph hard to follow. But in category theory, we can say that m is principle if and only if for every arrow x from A to A, and for every arrow y from A to B, the diagram to the right commutes.”

COMMENT: First of all, it’s “principal” morphism not “principle” morphism. Second of all, you fucked up the picture whose arrows unfortunately go the wrong direction from your definition.“I was just perusing my stats on sitemeter – and to my amazement, I discovered that Good Math/Bad Math had its 2 millionth view this weekend. 2 million pageviews! I never dreamt that this little blog would ever see a number like that.”

COMMENT: Mark, I’m surprised you can count that high.What really gets me is that someone who is so OBVIOUSLY mathematically ILLITERATE (as is obvious to anyone with a real math degree) is able to attract such a cult-like following and praise on the internet. Friggin unbelievable.

darin

Mark C. Chu-CarrollFor those who don’t remember him, Darin Brown is an AIDS denialist, who’s pissed off at pretty much everyone at ScienceBlogs for debunking his incredibly idiotic anti-HIV arguments. If you want to see both why Darin dislikes me so much, and the brilliance of his allegedly qualified mathematical opinions, take a look here.

While criticizing me for things like mis-spelling “principal” (which I admit was a stupid mistake…), Darin makes some truly astonishingly innumerate claims. One of his major arguments muddles the difference between aggregate

correlations and individual correlations.

Specifically, he wants to dispute HIV as the cause of AIDS.

Now, if we look at statistics, we can see that in the population of HIV+ individuals

as a whole, increased viral load correlates amazingly well with onset of symptoms. But taken on an individual level, the viral load isn’t a terrible good predictor of the onset of symptoms for an individual. According to Darin, that indicates that the correlation between viral loadin aggregatemust be false.To anyone with the least clue about statistics, this is obvious nonsense. The fact that he’s willing to stand by that argument indicates one of two things. Either he doesn’t understand anything about statistics, or else he’s a deliberate liar. In either case, he’s not someone I take seriously.

Yuri_KYADI have never studied this subject before, but I find it interesting. Would it be at all possible for you to show a quick proof of why the char(F) is either 0 or a prime number?

Jonathan Vos PostSketch of proof.

Consider a field F with multiplicative identity 1. Now consider the numbers 2=1+1, 3=1+1+1, 4=1+1+1+1, and so on.

Either these numbers are all different, in which case F is said to have characteristic 0, or two of them will be equal.

In the latter case, it is straightforward to show that, for some number p, we have 1+1+1…+1 (p times) = 0. If p is chosen to be as small as possible, then p will always turn out to be a prime, and we say that F has characteristic p.

For p a prime, the finite field GF(p^n) has characteristic p.