- #1

Kindayr

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When I was playing around with it, I defined a binary operation that basically, now that I've refined it, takes any two positive rationals, takes the polynomials associated with them in Z[X], multiply those polynomials, then take the inverse of my original map back to the positive rationals.

That is, I defined a map [itex]\phi :\mathbb{Q}^{+}\to\mathbb{Z}[X][/itex] such that if [itex]a\in\mathbb{Q}^{+}[/itex], then there exists [itex]k\in\mathbb{N}[/itex] and [itex]e_{0},e_{1},\dots ,e_{k}\in\mathbb{Z}[/itex] such that [itex]a=\prod^{k}_{i=0} p_{i}^{e_{i}}[/itex] and thus [itex]\phi (a)=(e_{0},e_{1},\dots,e_{k},0,0,\dots)[/itex]. Its easy to show that this an isomorphism and that [itex]\phi(ab)=\phi(a) +\phi(b)[/itex].

I then defined a binary operation [itex]\oplus :\mathbb{Q}^{+}\times\mathbb{Q}^{+}\to \mathbb{Q}^{+}[/itex] such that [itex]a\oplus b=\phi^{-1} [\phi(a) \cdot \phi(b)][/itex].

It is then easy to show that [itex](\mathbb{Q}^{+},\cdot,\oplus,1,2)[/itex] is an integral domain and that [itex]\phi[/itex] is an isomorphism of integral domains.

I was wondering now if I could find the field of fractions for the integral domain I just defined. What field would I get out of it?

Anyone have an answer?

Its a little late to work it out but I'll see where I get and post it up here :)