Hello. I have a few questions about Legendre and extended quadratic legendre symbols.

I understand the definition of the legendre symbol, but when it comes to the extended or Jacobi symbol I get a little confused. It says that:

For arbitrary integers n which factor as

n=(2^e(0))*(P(1)^e(1))*....*(P(k)^e(k)) the jacobi symbol is defined as

(b/n)=[(b/P(1))^e(1)]*[(b/P(2))^e(2)]....*[(b/P(k))^e(k)].

In our notation, those e's are sometimes just positive numbers, and other times they are the identity element, but it doesn't define it here. If it is the identity element, could someone elaborate where they come into play?

For an example, is it correct to say:

if n=pq where p and q are distinct primes, and (a/n)=-1,

then (a/n)=(a/p)*(a/q)=-1?

Sorry for the cramped notation, I've seen people use the proper notation on these forums, but I don't know how to do it

Thanks in advance.