How can I tell if a prepositional phrase is a complement to a noun or a modifier? And how are these two different?

In the NP "mines in wartime", "in wartime" modifies the head "mines".

that nice tall man from Canada whom you met

"from Canada" modifies "man".

But why is the prepositional phrase "of Education" a complement to "Ministry" in "the Ministry of Education"?

How can I tell if a prepositional phrase is a complement to a noun or a modifier? And how are these two different?