Is Every Rootless Polynomial Over a Finite Field Prime?

[wxrebecca]

How to prove that a polynomial of degree 2 or 3 over a filed F is a prime polynomial if and only if the polynomial does not have a root in F?

and i can't think of an example of polynomial of degree 4 over a field F that has no root in F but is not a prime polynomial.

it says each polynomial f(x) in F[x] determines a function from F to F by the rule c--> f(c). such a function is called a polynomial function from F to F. how to prove the different polynomials determine different functions when F is an finite field?


thanks for the help

Rebecca

 

[mathwonk]

the most basic result on roots of polynomials says that r is a root if and only if (x-r) is a factor.