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Wilson's theorem

Synonym: 
Wilson-Lagrange theorem
Type of Math Object: 
Theorem
Major Section: 
Reference

Mathematics Subject Classification

11A25 no label found37-01 no label found11-00 no label found

Comments

$p$ is a prime
IF AND ONLY IF
$(p-1)!\equiv -1\pmod{p}$

now tell me, isn't that great?
f
G ---------> H
\ ^ G
p \ /_ ----- ~ f(G)
\ / f ker f
Y /
G/ker f

Yes, the entry contains only half of the Wilson's theorem.

Wansn't this supposed to be a corection? As the person who posted it keeps on saying when people post things like this, if you leave it as a post, then it will hang around under the entry after the entry has been fixed and confuse people ;)

I think you can see the date of the post, I even think that the correction system wasn't even fully functional by then, that's why the post
f
G -----> H G
p \ /_ ----- ~ f(G)
\ / f ker f
G/ker f

Never mind, then. I assumed that since someone posted to this today, that the original post must have been somewhat recent. In that case, this only goes to show how right Drini is about stuff hanging around --- even experienced users can get confused years later. I guess this goes to prove the point that it would one day be nice to have a way for authors to make posts under their entries invisible or mnove them to a discussion area.

The entry "Wilson's theorem" is now completed and ready.

I perceive that not all people here believe the negative primes (-2, -3, -5 and so on). But in number-theoretical sense, they are as good as the positive primes, since both have exactly same divisibily properties. And ideal-theoretically, p and -p generate the same principal ideal, as always do the associates. In the other algebraic number fields than Q we, in fact, can not say that some of the associated primes is "better" or "more correct" than some else.
The (rational) prime defined in PM may be negative equally as positive.
Jussi

Its not an issue of "better" or "more correct": its an issue of how the primes are usually defined. It was my understanding the normal definition excludes negatives; and you have to explicitly include negatives to consider them. I'll put a clarification in the entry, but can anyone else comment on conventions?

apk

To a vast majority of people, a prime number refers to the positive version, so I'd suspect that that's the best way to define it in an entry. Number-theorists commonly, by an abuse of terminology, use the term "prime" to semi-interchangeably mean either the prime number itself, or the principal ideal generated by that prime (or even, some times, the collection of prime ideals above a given prime in a field extension).

One drawback to including the negative versions is that unique factorization does not hold, at least not without an additional disclaimer. For example, 6=2*3 and 6=(-2)*(-3) would be two "different" factorizations if we took our list of primes ti include the negative ones. On the plus side, the corrected version of the statement, that we have a unique factorization into *ideals* (since 2 and -2 generate the same ideal) motivates one of the cornerstones of algebraic number theory, i.e. that we *always* have unique factorization of ideals into prime ideals in number fields.

So I'd suggest leaving the definition in terms of positive integers, but perhaps noting that there are benefits obtained by including this more general definition.

Cam

I think my train of thought was perhaps too limited in that last post. There is one major logistical reason why we should only include positive numbers in the principal definition of a prime number: Very many definitions and theorems would have to be (presumably slightly) re-worded to incorporate a definition of "prime number" that included the negative primes. For example, if -2 were a prime, it would no longer be true that there is a finite field of cardinality p for every prime number p. as there are no fields with -2 elements. I suspect it would be easy to come up with hundreds of other examples.

I would leave the definition as it is, i.e. prime numbers are positive. The reason is the one mathcam pointed out, if we allow negative primes, then the unique factorization theorem is akward to state.

alvaro

It really isn't that much more akward (not that I'm suggesting changing it, really). You simply say that if

x = p_1 ... p_r = q_1 ... q_s

then r = s and up to ordering, p_i = u q_i, where u is a unit.
--
"It's John Ashcroft's vision of hell. It's a Tom Waits song."
Simon - mhm27x5

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