Re: On The Fundamental Theorem of Arithmetic and Why it Breaks Down for the Algebraics

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)
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Arturo Magidin
magidin-at-member-ams-org

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