Up until now, I've mainly worked with the polynomial ring $\mathbb{R}[x_1,...,x_n]$ or the corresponding field of fractions. But I can't think of an example of a field of functions that contain non-polynomial functions. What examples are there of a field of functions that contain the set of real rational functions and the natural logarithm function $ln(x)$? Thanks!
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You can definitely study the field of functions $\mathbb{R}(x_1,\ldots, x_n)=\text{Frac}(\mathbb{R}[x_1,\ldots, x_n]).$ This is the space of rational functions in the variables $x_1,\ldots, x_n$. More abstractly, for any integral domain $A$, one can study $A[x_1,\ldots, x_n]$ and $A(x_1,\ldots, x_n)$.
As far as an example of a field containing $\log$ or something like that, we can take $\Omega=\mathbb{C}\setminus [0,\infty)$ and study $$\mathcal{M}(\Omega)=\{\text{meromorphic functions on}\:\Omega\}.$$ Among these functions is $\log(z)$ where we use the branch cut $\text{arg}(z)\in (0,2\pi)$. It can be shown that the ring $\mathcal{M}(\Omega)$ is actually a field, containing $\log$.
answered Jan 4, 2019 at 17:31
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$\begingroup$ That makes sense, would the field of meromorphic functions also include sin(z) and cos(z)? (I haven't taken complex analysis yet so my knowledge of this stuff is a little shaky). Is the set of holomorphic functions on $\Omega$ also a field? $\endgroup$ Jan 4, 2019 at 17:46
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$\begingroup$ $\sin$ and $\cos$ would also be included. The holomorphic functions are not a field, because if you take $1/(z-i)$ (for instance), we can see that it isn't holomorphic: it has a singularity at $z=i$. $\endgroup$ Jan 4, 2019 at 17:47