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From: Fred on 21 Jul 2010 11:48
On Jul 21, 6:32 am, "G. A. Edgar" <ed...(a)math.ohio-state.edu.invalid>
wrote:

> A remark on the definition of "algebraic function" ...

> For example, we do not want to allow: f(x) = sqrt(1-x^2) for
> x rational in (-1,1) and f(x) = -sqrt(1-x^2) for x irrational
> in (-1,1).

Why don't we want to allow that? Even more so, don't we want to allow |
x| to be algebraic since it satisfies the polynomial y^2 - x^2?

--Fred
From: G. A. Edgar on 21 Jul 2010 21:44
In article
<b71bab5e-1793-4ca2-b822-d6661fb37071(a)g19g2000yqc.googlegroups.com>,
Fred <f.richman(a)comcast.net> wrote:

> On Jul 21, 6:32�am, "G. A. Edgar" <ed...(a)math.ohio-state.edu.invalid>
> wrote:
>
> > A remark on the definition of "algebraic function" ...
>
> > For example, we do not want to allow: f(x) = sqrt(1-x^2) for
> > x rational in (-1,1) and f(x) = -sqrt(1-x^2) for x irrational
> > in (-1,1).
>
> Why don't we want to allow that? Even more so, don't we want to allow |
> x| to be algebraic since it satisfies the polynomial y^2 - x^2?
>
> --Fred

No, we don't. At least I don't. But then I would start with an
analytic function defined on a domain of the complex plane. Or
maybe defined on a Riemann surface.

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
From: Fred on 22 Jul 2010 22:08
On Jul 21, 9:44 pm, "G. A. Edgar" <ed...(a)math.ohio-state.edu.invalid>
wrote:
> In article
> <b71bab5e-1793-4ca2-b822-d6661fb37...(a)g19g2000yqc.googlegroups.com>,
>
> Fred<f.rich...(a)comcast.net> wrote:
> > On Jul 21, 6:32 am, "G. A. Edgar" <ed...(a)math.ohio-state.edu.invalid>
> > wrote:
>
> > > A remark on the definition of "algebraic function" ...
>
> > > For example, we do not want to allow: f(x) = sqrt(1-x^2) for
> > > x rational in (-1,1) and f(x) = -sqrt(1-x^2) for x irrational
> > > in (-1,1).
>
> > Why don't we want to allow that? Even more so, don't we want to
> > allow |x| to be algebraic since it satisfies the polynomial
> > y^2 - x^2?
>
> > --Fred
>
> No, we don't.  At least I don't.  But then I would start with an
> analytic function defined on a domain of the complex plane.  Or
> maybe defined on a Riemann surface.
>
> --G. A. Edgar

I guess it all depends on where one is coming from. My motivation is
the definition of "algebraic function" given in most calculus books.
They clearly consider |x| to be algebraic because it is sqrt x^2.

--Fred
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