While reading about symbolic integration I encountered some concepts of Differential Algebra. I do not know much of D.A and Fields in general also I have encountered as an extension of Rings. I haven't done a full course on Field Theory. My question is can you give me an example of elementary field extension as described in this image. Kindly be a bit elaborate, it would help me understand the concept. :)
Start with the base field $\,\Bbb E_0 = \Bbb C(x) =\,$ the field of rational functions in $\,x.\,$ Now adjoin to $\Bbb E_0$ any $\,\theta_1$ being a logarithm $\log(f),\,$ or an exponential $\,\exp(f)\,$ or algebraic function (e.g. $\sqrt{f})$ of any $\,f\in \Bbb E_0 = \Bbb C(x),\, $ to obtain $\,\Bbb E_1 = \Bbb E_0(\theta_1).\,$ Continue in the same way, by adjoining logs, exps, or algebraic functions of elements in prior differential fields until one obtains the required differential field (e.g. a field containing the integrand).
For example, to construct a differential field containing $\,\log(x+\exp(\sqrt{x}))$ we can successively adjoin $\,\theta_1 = \sqrt{x},\,\ \theta_2 = \exp(\sqrt{x}),\,\ \theta_3 = \log(x+\exp(\sqrt{x}))\,$ to obtain $\,\Bbb E_3 = \Bbb C(x,\theta_1,\theta_2,\theta_3).$