I'm newer to abstract algebra, and recently I came back to the concept of a field after finding rational functions form a field.
But, what I notice is that rational functions (and algebraic functions) are closed under composition.
If I equip a field with $+, \cdot, -, /,$ then what abstraction can I add to include $\circ,$ or function composition?
How does one generalize the idea of functional composition to include it to a field, and does such an object have a name?
You can require the existence of a composition operation $\circ$ on your field, thus creating a composition field. Any field may be considered as a composition field with the "trivial" composition $f\circ g=0$, or the "constant" composition $f\circ g=f$. For any interesting composition rules, however, you may have to consider more specialized fields. See more in the link provided by healynr.