Changes http://wiki.axiom-developer.org/DistributedMultivariatePolynomial/diff -- William wrote:
> $R[x]$ is a subring of $Q(R)[x]$ where $Q(R)$ is the quotient > field of $R$. I find your notation a little confusing. In Axiom notation I think you mean, for example: $R[x]$ = POLY INT then: $Q(R)[x]$ = POLY FRAC INT Is that right? If so, then I agree that: > '1/x' is in neither of these polynomial rings, But I do not agree with your continuation of the statement: > (((No matter what $R$ is!))). This is a FACT of mathematics, > the meaning of what an indeterminate x over a ring or > field is. 1/x ONLY lives in $Q(R[x]) = Q(Q(R)[x]$, where > $(1/x)\times x = 1$. By $Q(R[x])$ I suppose you mean, for example: $Q(R[x])$ = FRAC POLY INT I don't understand why you write $Q(Q(R)[x]$ meaning for example: $Q(Q(R)[x])$ = FRAC FRAC POLY INT but certainly '1/x' does "live" in these domains, if by "live" you mean that a suitable function '/' can be found by a selection from these domains. However '1/x' is also lives in other polynomial rings such as: $Q(R[x])[x]$ = DMP(![x], FRAC POLY INT) where $(1/x)\times x \ne 1$. $Q(R[x])[x]$ satisfies all of the axioms of a ring and the 'x' in $Q(...)[x]$ remains transcendent over the 'x' in $R[x]$. You can see that a valid selection for the function '/' exists in this domain by doing:: )sh DMP([x], FRAC POLY INT) It is convenient that you choose to write $\times$ above because we need to distinquish two different multiplications: the $\times$ in DMP and the '*' in 'POLY INT'. It is only when we coerce $Q(R[x])[x]$ into $Q(R[x])$, which means that we map the $\times$ and '/' of 'DMP(![x], FRAC POLY INT)' into the $*$ and '/' of 'FRAC POLY INT' that we can equate these two uses of $x$. In making this claim I have not violated any FACT of mathematics. -- forwarded from http://wiki.axiom-developer.org/[EMAIL PROTECTED] _______________________________________________ Axiom-developer mailing list [email protected] http://lists.nongnu.org/mailman/listinfo/axiom-developer
