Although inspired by my question on math.SE https://math.stackexchange.com/q/1902190/214353 this is not a crosspost. What happened is that after an answer and some comments I realized more clearly what did I want there.
Suppose given a field extension $k\subset K$ and I want to extend $K$ to an $L$ by adding to $K$ a new element $t$ with certain properties (algebraic or maybe transcendental). This may be viewed either as taking quotient $K[t]/P$ of $K[t]$ by the ideal $P$ defining these properties, or - if these properties do not depend on $K$ - as $K\otimes_kA$ for the similar quotient $A=k[t]/p$ of $k[t]$.
What I "don't like" here is that automorphisms of $K$ over $k$ are not taken into account at all. In particular, it is assumed that they all fix $t$, so that, in the second version above, the fixed field of automorphisms of $L$ over $k$ contains the whole $A\subset L$. Which might be sometimes desirable, but sometimes not.
So my question is - is appropriate formalism known to deal with extensions having prescribed not generally trivial actions of automorphisms of $K$ on added elements? Say in the example above I pick an automorphism $\sigma\in\operatorname{Gal}(K/k)$ and want $t^\sigma$ to be something definite but different from $t$.
I understand that this might be simply incorporated into the defining relations for the ideal $P$ and just means that the second option described is not available. Still, I have feeling that there is more precise way to act, somehow separating general algebraic relations on one hand and Galois actions on the second. I want to know what (if any) such precise way may be.
Also I would like to know what is the dual geometric counterpart of the above. The setup must be something like coverings $X\to Y\to Z$, and one asks for describing how deck transformations of $X\to Y$ interact with those of $Y\to Z$. Are there techniques dealing with such situations?
Finally, a specific example related to that math.SE question, to illustrate the above and also to make the question less foggy.
Consider the field ${\mathbb C}(t)$ of rational functions with complex coefficients, and let $\sigma:{\mathbb C}(t)\to{\mathbb C}(t)$ be given by $$ \sigma\left(\frac{f(t)}{g(t)}\right)=\frac{\bar f(-\frac1t)}{\bar g(-\frac1t)}, $$ where $f$ and $g$ are polynomials, whereas $\bar f$ and $\bar g$ are obtained from them by replacing coefficients with their complex conjugates.
Thus defining $\bar r:=\sigma(r)$ for a rational function $r\in\mathbb C(t)$ extends conjugation from $\mathbb C$ to $\mathbb C\subset\mathbb C(t)$ in a certain way, and seemingly strange thing happens: although $\mathbb C\subset\mathbb C(t)$ is of course a transcendental extension, if we regard complex conjugation as part of the algebraic signature - that is, view this extension as a homomorphism of rings-with-an-additional-unary-operation - it seems that this actually becomes an algebraic rather than transcendental extension, since with the above extension of conjugation $\mathbb C(t)$ becomes the extension of $\mathbb C$ by a root of the equation $$ t\bar t=-1. $$
So what kind of algebraic extension is that? Is there a theory treating such transcendental extensions as algebraic ones on account of incorporating automorphisms into the algebraic signature as additional unary operations?
Let me add that somehow maybe the Picard-Vessiot theory is something of an infinitesimal version of what I am up to.
