Hyperreal field extension

Question

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).

Answer 1

I don't think it would be enough to adjoin one infinitestimal. To construct the hyperreals via an ultrapower requires the Axiom of Choice, but you could certainly adjoin one infinitesimal without Choice.

Answer 2

Briefly, one way to approach this question would be to try to construct alternative models of the hyperreals via the Compactness Theorem that either satisfy or fail to satisfy the properties you want. The Enderton text, A Mathematical Introduction to Logic, uses this kind of construction and has a very rigorous treatment of nonstandard analysis that you can use as a guide.

Since I only have 44 minutes left on this question, it could easily contain errors so I'm retagging your question to include logic and set theory so it can be properly checked.

Answer 3

There isn't any trouble constructing $\mathbb{R}(\epsilon)$ as a formally real field that is isomorphic (as a field) to $\mathbb{R}(x)$, and with $\epsilon$ a positive infinitesimal. (it may be easier to see the ordering by writing a rational function as a formal Laurent series in $\epsilon$ about 0)

It's easy to see that this is not a hyperreal field: it only has finite powers of $\epsilon$. In particular, if there were a positive transfinite positive integer $H$, then there would be an element $\epsilon^H$ that is smaller than every power of $\epsilon$ appearing in $\mathbb{R}(\epsilon)$.

($\mathbb{R}(\epsilon)$ also fails to have a square root of $\epsilon$. This can be fixed by passing to its real closure, but that would still fail the above property)

If you tried adjoining a second infinitesimal, a similar argument proves it can't be hyperreal.