Shamir Secret Sharing Scheme - some doubts not answered earlier my first question on here.

I've gone through some examples of Shamirs Secret Sharing Scheme, but I have 2 rather  pressing doubts -

i. Why do we choose the co-efficients of the polynomial from the set {1,..., p-1}?  

ii. Why should we have p (the choice of prime) > n (total number of parties involved in the secret sharing scheme). I am able to reason why p > k but not the above!

Thanks very much in advance.
You need a finite algebraic object, resulting in exact arithmetic and exact recovery of the secret. So infinite fields such as $\mathbb{C,~R}$ are not suitable.

Since the scheme is based on polynomial interpolation, you need two operations, i.e., a field. It can be a prime field $\mathbb{F}_p$ or an extension field $\mathbb{F}_{p^m}.$

Since each user is assigned their secret by specifying $y_i$ at point $x_i,$ you need at least $n+1$ points, the value at zero of the polynomial normally being used for specifying the secret.