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Math Help - Polynomials of degree n over Zp

  1. #1
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    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.
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by page929 View Post
    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 \mathbb{Z}_{3}

    Hint 1: A degree 2 polynomial has the form

    P(x)=a_2x^2+a_1x+a_0 where a_i \in \mathbb{Z}_3 and we know that a_2 \not\equiv 0 \text{mod} 3

    Since our finite field has only 3 equivalence classes [0],[1] \text{ and } [2] there are two choices for a_2. Since there is no restriction on a_1 or a_0(they do not affect the degree of the polynomial) they each have three choices. This gives
    2\cdot 3 \cdot 3=18

    See if you can generalize this.
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  3. #3
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    Thank you. I believe I have it now.
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