I am studying JCF and RCF from Dummit & Foote. I can say what are all the differences between them, either in computations or in definitions and constituents. Could someone clarify this to me please?
This is my first time to study these concepts deeply.
When a minimal polynomial does not factorize completely, JCF doesn't work. For example, $x^2 − 2$ cannot be factorized in $\mathbb{Q}[x]$ since $\sqrt{2} \notin \mathbb{Q}$. However, the RCF works in this case.
You may think of JCF turns a finite dimensional vector space $V$ as a direct sum of as many cyclic subspaces as possible.
While RCF turns $V$ as a direct sum of as few cyclic subspaces as possible.
Since you are reading D&F, you may figure out that both JCF and RCF are just special cases of the Structure theorem for finitely generated modules over a PID.