I would appreciate it if you guys would help me to understand the idea behind these two concepts: Gauge field and Yang-Mills theory.
What I think I understand is: Suppose we have a Lagrangian that describes a field itself, $L$. The Lagrangian can be, for example, globally invariant under $U(1)$ transformation. So, we now impose the Lagrangian needs to be invariant locally under $U(1)$, making the parameter $\alpha$ on $e^{i \alpha}$ dependent on position: $e^{i\alpha(x)}$.
Doing this, we need to: 1) add a gauge free field term on the Lagrangian, and 2) define a new derivative: the covariant derivative.
The point now is that I don't understand what we gain with this! Imposing the Lagrangian to be invariant locally results in exactly what?
