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sci.math > {group theory} positive rationals under multiplication = polynomials under addition
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Sniz Pilbor   Dec 5 2003, 8:51 pm     show options
Newsgroups: sci.math
From: (Sniz Pilbor) - Find messages by this author
Date: 5 Dec 2003 17:51:51 -0800
Local: Fri,Dec 5 2003 8:51 pm
Subject: {group theory} positive rationals under multiplication = polynomials under addition
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     Been reading "The Theory of Groups: An Introduction (2nd
Edition)", a huge and advanced tome by Rotman.  In one of the
exercises, he casually mentions that Z[x], the polynomials with
integer coefficients, considered as a group under addition, is
isomorphic to the positive rationals considered as a group under
multiplication.  The exercise was to prove this by exhibiting such an
isomorphism explicitly, and I solved it thus:

- = (p1^k1) * (p2^k2) * (p3^k3) * ...

where p1,p2,... are the prime numbers and k1,k2,... are integers
which, by the fundamental theorem of arithmetic, exist and are unique.
 (They are not necessarily nonnegative, as for example 1/2 = 2^(-1))

Therefore, given a positive rational q, we are also given k1,k2,k3,...
and we use these to construct
i(q) = k1*(x^0) + k2*(x^1) + ... + kn(x^(n-1)) + ...,
a function from the positive rationals to Z[x].
It's not hard to show that i is an isomorphism.

As some examples:
i(0)   = the zero polynomial
i(1/2) = -1 (the polynomial)
i(3/5) = x - x^2
i(3.14159265) = -8 + 2x - 7x^2 + x^3 + x^30 + x^991

This all seems incredibly interesting and profound.  It means, for
example, that we can sensibly refer to the "zeros", the "degree", the
"maxima", the "derivative", etc. of a positive rational, or of "the
value at x=5 of" the same.  Given a rational, we can figure it's
isomorphic image among Z[x], differentiate this and go back to Q to
obtain a new rational.. like so:
d/dx 3.14159265 = i^-1(2 - 14x + 3x^2 + 30x^29 + 991x^990)
                = (2^2)*(5^3)*(113^30)*(7841^991)/(5^14)

It seems there is no limit to the amount of new operations we can
invent using this isomorphism, the above are just a few examples.  We
could speak of "irreducible" rationals, "the minimum rational" of an
algebraic (ie, the rational image of its minimal polynomial), etc.
etc. etc.

Unfortunately, the book only briefly mentioned this fascinating
isomorphism in the exercise, and that's the end.  Now I'm horribly
filled with unsated curiosity.  Has anyone studied this isomorphism in
more detail?  Are there any papers on it?

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