I am reading a textbook about group representation and it says:
A field $\mathbb{F}$ is called a splitting field for a finite group $G$ if for every irreducible representation $\rho$ of $G$ the only intertwining operator (a morphism) between $\rho$ and itself are the scalar multiples $cI$ of the identity map $I$: $V_\rho \rightarrow V_\rho$, where $V_\rho$ is the vector space over $\mathbb{F}$.
However, when I read the splitting field in Wiki, it says something about polynomial:
https://en.wikipedia.org/wiki/Splitting_field
My question is: are both "splitting field" the same thing? I am really confused about both definitions.
There is a more general notion of splitting field for a finite-dimensional algebra and it integrates the two usages. So they are essentially the same. See Proposition 7.12 and Definition 8.2 in [T. Y. Lam, A First Course in Noncommutative Rings (2001)].