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I have looked (a bit) at questions like finding the number of roots of $x^n =1$ over a finite field.

Now I would like to understand how to solve polynomial equations over finite fields. From what I understand the solution to the quadratic equation has the same solution formula. That means that it all comes down to finding square roots.

I know that there are formulas for solving higher degree equations, but I am wondering if some of these things look different in some way over a finite field.

I am asking for a reference that specifically deals with solving polynomial equations over finite fields. It would be nice if the reference is accessible to an advanced undergraduate student.

user87952
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    Be a little careful the quadratic equation does not hold in fields of characteristic 2 since it requires you to divide by 2. That is you need 2 to be a unit. – John Machacek Jun 13 '14 at 00:26
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    That the general quintic with coefficients in $\mathbb C$ cannot be solved with radicals is a frequently repeated mantra. But are there fields in which the answer is different? I expect there are many to whom the answer is a triviality, but I don't know it. – Michael Hardy Jun 13 '14 at 03:29

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Solving polynomials in one variable over finite fields is substantially easier than solving polynomials in general. To find out if $f(x) = 0$ has any roots over $\mathbb{F}_q$ you just need to compute $\gcd(f(x), x^q - x)$ using the Euclidean algorithm, since the roots of $x^q - x$ are precisely the elements of $\mathbb{F}_q$. An elaboration of this idea leads to an efficient algorithm for not only solving but even factoring polynomials over finite fields; see Berlekamp's algorithm for details.

Qiaochu Yuan
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