Let x be a negative rational number, then x is not in the subgroup generated by all positive primes which has index 2. If x=a/b is a positive rational number such that the number of prime factors of a plus the number of prime factors of b is odd, then x is not in the subgroup generated by all negative primes which also has index 2. But i am stuck with the case where the number of prime factors of a plus the number of prime factors of b is even.

Does anybody know how to work that out?

Thankful for any help

Banach