Thread: Jacobi Symbol Properties

This is a theorem about Jacobi symbols in my textbook:
Let n and m be ODD and positive. Then (a/nm)=(a/n)(a/m) and (ab/n)=(a/n)(b/n)
Moreover,
(i) If gcd(a,n)=1, then ( ) = 1 = ( )
(ii) If gcd(ab,nm)=1, then ( )=(a/n)
=====================================

(i) is easy and follows from the definition, but how can we prove (ii)? My textbook stated the theorem without proof and just says the proofs are easy, but I have no idea why (ii) is true.

Any help is appreciated!

Originally Posted by kingwinner

This is a theorem about Jacobi symbols in my textbook:
Let n and m be ODD and positive. Then (a/nm)=(a/n)(a/m) and (ab/n)=(a/n)(b/n)
Moreover,
(i) If gcd(a,n)=1, then ( ) = 1 = ( )
(ii) If gcd(ab,nm)=1, then ( )=(a/n)
=====================================

(i) is easy and follows from the definition, but how can we prove (ii)? My textbook stated the theorem without proof and just says the proofs are easy, but I have no idea why (ii) is true.

Any help is appreciated!

Think it is because...





Was it supposed to be ? That was a bit unclear

Originally Posted by Deadstar

Think it is because...

Why (b^2/nm^2)=1? How do you know that gcd(b,nm^2)=1?
Why (a/m^2)=1? How do you know that gcd(a,m)=1?

Was it supposed to be ?

No.

Thanks for explaining!

Originally Posted by kingwinner

Why (b^2/nm^2)=1? How do you know that gcd(b,nm^2)=1?
Why (a/m^2)=1? How do you know that gcd(a,m)=1?

Since , we have that

The other case follows similarly.