Final answer:
Yes, the product of two primitive roots mod an odd prime p can also be a primitive root mod p. This is because both primitive roots have orders p-1, and their product can maintain this order.
Step-by-step explanation:
The question is about whether the product of two primitive roots mod p, where p is an odd prime, can also be a primitive root mod p. Suppose g and h are primitive roots mod p. By definition, a primitive root mod p is an integer x such that every number coprime to p is congruent to a power of x, modulo p. In other words, x generates the multiplicative group of integers modulo p.
The order of a primitive root mod p is p-1. Therefore, g and h both have orders p-1. The product g*h can be a primitive root if and only if its order is also p-1. This is possible if g and h are chosen such that their product does not reduce the order of the resulting number.
Since the set of exponents that g and h can be raised to in order to generate all residues mod p are complete and distinct (modulo p-1), their product will also have a complete set of exponents when taken modulo p. Therefore, the product g*h can be a primitive root mod p.