My questions is...
How many polynomials are there of degree n over Zp? Where p is a prime number and n is a positive integer.
I don't fully understand degrees.
Thanks, in advance, for any help.
How about we start with an easier question! How many polynomials of degree 2 are there over $\displaystyle \mathbb{Z}_{3}$
Hint 1: A degree 2 polynomial has the form
$\displaystyle P(x)=a_2x^2+a_1x+a_0$ where $\displaystyle a_i \in \mathbb{Z}_3$ and we know that $\displaystyle a_2 \not\equiv 0 \text{mod} 3$
Since our finite field has only 3 equivalence classes $\displaystyle [0],[1] \text{ and } [2]$ there are two choices for $\displaystyle a_2$. Since there is no restriction on $\displaystyle a_1$ or $\displaystyle a_0$(they do not affect the degree of the polynomial) they each have three choices. This gives
$\displaystyle 2\cdot 3 \cdot 3=18$
See if you can generalize this.