# Polynomials of degree n over Zp

• Feb 8th 2011, 07:01 AM
page929
Polynomials of degree n over Zp
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.
• Feb 8th 2011, 07:48 AM
TheEmptySet
Quote:

Originally Posted by page929
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.
• Feb 8th 2011, 08:23 AM
page929
Thank you. I believe I have it now.