primitive root of residue modulo p I was trying to prove that for the set $\{1,2,....,p-1\}$ modulo p there are exactly $\phi(p-1)$ generators.Here p is  prime.Also the operation is multiplication.
My Try:
So I first assumed that if there exists a generator, then from the set $\{a^1,a^2,...,a^{p-1}\}$ all those powers which are co prime to $p-1$ are also generators.But i am having difficulty in proving the existence of such an $a$. If anyone can help it would be great . Thanks.It would be better if it avoids fundamental theorem of algebra. 

Assuming $p$ is prime then $\mathbb{Z}/p\mathbb{Z}$ is the finite field $\mathbb{F}_p$ and the set you are interested in is the multiplicative group $\mathbb{F}_p^*$.  In this context what you are looking for is a proof that the multiplicative group is cyclic. And with this formulation you can find a lot of answers, for example this collection of proofs: https://mathoverflow.net/questions/54735/collecting-proofs-that-finite-multiplicative-subgroups-of-fields-are-cyclic
