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In non-standard analysis, assuming the continuum hypothesis, the field of hyperreals $\mathbb{R}^*$ is a field extension of $\mathbb{R}$. What can you say about this field extension?

Is it algebraic? Probably not, right? Transcendental? Normal? Finitely generated? Separable?

For instance, I was thinking: Would it be enough to adjoin an infinitesimal to $\mathbb{R}$ and get $\mathbb{R}^*$? The axioms of hyperreals in Keisler's Foundations of Infinitesimal Calculus seem to suggest so, but I'm not sure.

I don't know how to approach this question, since infinitesimals and such things don't "result" from polynomials (like, say, complex numbers do).

Is $\mathbb{R}^*$ maybe isomorphic to the field of fractions of some polynomial ring? (This occured to me after thinking about $\mathbb{R}(\epsilon)$ and such things, where $\epsilon$ is an infinitesimal).

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    I just realised it's quite easy to show it isn't algebraic. But what about the rest?2012-11-10
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    There actually exists no single field of hyperreals. Different fields of hyperreals may fail to be isomorphic.2012-11-10
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    Let's assume the continuum hypothesis, then. If I remember correctly, that implies that all are isomorphic, right?2012-11-10
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    @FFP Yes.${}{}$2012-11-10
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    Well, exactly what should $\epsilon$ satisfy? How is (an) $\Bbb R^*$ constructed?2012-11-10
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    @Berci I guess it would be an arbitrary infinitesimal already in $\mathbb{R}^*$. Don't think of it as constructing $\mathbb{R}^*$ "from scratch", but whether $\mathbb{R}(\epsilon)$ would "fill up" all of $\mathbb{R}^*$, if that makes sense. Just to give $\epsilon$ some meaning before hand.2012-11-14
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    Closely related: http://math.stackexchange.com/questions/167250/is-there-more-than-one-infinitessimal-among-the-hyperreal-numbers/2012-11-21

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