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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).
--
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"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================
Arturo Magidin
magidin-at-member-ams-org
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