3NF Decomposition and NP-Completeness

I have a question about the computational complexity of the algorithm that allows us to decompose a schema into 3NF. My book says that the decisional problem of telling if a schema complies with the 3NF is NP-Complete. An NP problem is a decisional problem that can be solved in a polynomial time by a non-deterministic algorithm and verified in a polynomial time by a deterministic algorithm. An NP-Complete problem is a problem that is as hard to solve as the SAT problem. So, if I've understood correctly, if somebody gives me a schema and tells me wheater or not it is in 3NF I can verify it in polynomial time with a deterministic algorithm. But if somebody gives me the actual problem and not the decisional one by asking me: "Find a 3NF decomposition of my schema" I wouldn't be able to find a polynomial solution with a deterministic algorithm. I would be glad if somebody could tell me if there are any mistakes in my understanding. Thank you in advance!