|
|
In sci.math, Arturo Magidin
<magidin@xxxxxxxxxxxxxxxxx>
wrote
on Sun, 31 Dec 2006 14:14:57 +0000 (UTC)
<en8gl1$qso$1@xxxxxxxxxxxxxxxxxx>:
> In article <870j64-nsg.ln1@xxxxxxxxxxxxxxxxxxxxxxx>,
> The Ghost In The Machine <ewill@xxxxxxxxxxxxxxxxxxxxxxx> wrote:
>>I have a dumb question, and am not sure precisely how to
>>phrase it to Google.
>>
>>As everyone in this forum should know, for every positive
>>integer N, one can find a unique decomposition into primes
>>such that
>>
>>N = 2^e_2 * 3^e_3 * 5^e_5 * 7^e_7 * ... * p ^ e_p
>>
>>where e_2, e_3, e_5, etc. are nonnegative integers,
>>and p are special positive integers usually called primes,
>>divisible by only themselves and 1.
>>
>>Usually the zero exponents are omitted, so that one get
>>things such as
>>
>>33 = 3 * 11
>>42 = 2 * 3 * 7
>>54 = 2 * 3^3
>>65 = 5 * 13
>>109 = 109^1
>>etc.
>>
>>This is of course the Fundamental Theorem of Arithmetic,
>>proven long ago by either Euclid or Gauss. I'll admit to
>>not being familiar with Ernst Kummer's work but am curious
>>as to why this factorization fails entirely in the ring
>>of algebraic integers, which are, of course, those roots
>>(real or complex) for irreducible members of Z[x] whose
>>highest power term has a coefficient of +/- 1.
>
> In the full ring of algebraic integers, there are no "primes" (there
> are no irreducible elements). So you cannot have unique factorization
> into primes; in fact, you have NO factorization into irreducibles for
> any algebraic integer other than units (which have the empty
> factorization).
>
That's fine as far as it goes; the rest of it was
an attempt why. I wanted to know if that attempted
explanation was correct and/or if there was an alternate
explanation which makes more sense.
But you are correct; there are no primes in the algebraic
integers (though there are quite a few units). In the
integers, all factorizations are finite (a positive
integer is either 1, a prime, or the product of two smaller
positive integers, both of which must be at greater than 1
and at most half the number), and indeed, the Fundamental
Theorem is arguably why everyone chases after primes in
the first place.
However, in algebraics, one gets nasty things such as
7 = 7^(1/2) * 7^(1/4) * 7^(1/8) * 7^(1/16) * ...
where everything involved is a non-unit integer.
Apparently, a variant of infinite integer factorization
is what breaks the Fundamental Theorem of Arithmetic therein.
--
#191, ewill3@xxxxxxxxxxxxx
Windows Vista. Because it's time to refresh your hardware. Trust us.
--
Posted via a free Usenet account from http://www.teranews.com
|
|