[FOM] Historical queries

Alasdair Urquhart urquhart at cs.toronto.edu
Mon Nov 12 10:17:38 EST 2007

Joe Shipman poses the following query:

> X1: Fundamental Theorem of Algebra (both existence and uniqueness of
> prime factorizations; but since existence is so easy, I will accept a
> statement of uniqueness; the important question is who first recognized
> that there is something that needs proving here)

Hardy and Wright "Introduction to the Theory of Numbers, 4th Ed." has
in Section 1.3 the Fundamental Theorem of Arithmetic as Theorem 2,
proved as a corollary to Theorem 2, described as "Euclid's First Theorem."

"If p is prime, and p|ab, then p|a or p|b."

In a footnote to the Section, they make the following remarks:

"Theorem 3 is Euclid VII.30.  Theorem 2 does not seem to have been
stated before Gauss (Disquisitiones Arithmeticae Section 16).
It was, of course, familiar to earlier mathematicians; but
Gauss was the first to develop arithmetic as a systematic science."

I had a quick look through Euclid Book VII, and indeed can't see
any indication that Euclid proved anything of the sort.
Proposition VII.31 proves the existence of prime divisors, but
I can't see uniqueness anywhere, though Euclid's clumsy terminology
makes it hard to make out what he is saying in Book VII.

Alasdair Urquhart

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