Are there diagonalizability criteria for matrices with entries in different fields

Question

The problem came to my head a year ago when I'm learning linear algebra.In analysis especially numerical analysis and matrix analysis,the matrix algebra $M_n(C)$ is studied most frequently because $C$ is a field equipped with inner product(So Hilbert,we can do Gram-Schmidt algorithm on it),is a complete metric space,more important it is algebraically closed.Then we can prove that every normal matrix(The matrix $A$ which satisfies $A^HA = AA^H$) is diagonalizable.How about square matrices on normal field,such as quotient field $\mathbb{Q}$ or finite field $\mathbb{F}_{q^n}$(here $A^TA=AA^T$,$A^T$ means the transpose of $A$)?moreover would you give me an insight on how to search for the rules of diagonalizable matrix on common fields?