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

2. 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?
n=(2^e(0))*(P(1)^e(1))*....*(P(k)^e(k))
all of the P s are prime, so the e s are the powers of each prime in the prime factorisation of n.

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?
yes

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
If you know latex, you can access the displaymath enviroment by entering or clicking the button with $\Sigma$ on it. If you don't know latex, you can how other people made pretty equations by clicking on them.