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I am trying to show that if $f(x) \in \mathbb{F}_p[x]$ ($p$ prime) is a polynomial of degree $5$, then $f(x)$ is irreducible if and only if $gcd(f(x), x^{p^2}-x)=1$.

Suppose $f(x)$ is irreducible. Now $f(x) \neq c(x^{p^2}-x)$ where $c \in \mathbb{F}_p$ because $p^2 \neq 5$. If $x^{p^2}-x$ divides $f(x)$, then because $f(x)$ is irreducible, we must have $x^{p^2}-x = 1$ which is a contradiction. If $f(x)$ divides $x^{p^2}-x$, then I have read from Product of all monic irreducibles with degree dividing $n$ in $\mathbb{F}_{q}$? that $x^{p^2}-x$ is the product of all monic irreducible polynomials with degree dividing $2$. This means $f(x)$ must equal (up to multiplication by a unit) some monic irreducible polynomial with degree dividing $2$ which is a contradiction as $f(x)$ has degree $5$. Is this logic correct?

For the other direction where we suppose that $gcd(f(x), x^{p^2}-x)=1$, I would appreciate any guidance on how to proceed.

Nick
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    $gcd(a,b) = 1$ does not imply that $a | b$ or $b | a$. Rather, it implies there is no $c \ne 1$ such that $c | a$ and $c | b$. The fact that every irreducible polynomial of degree 2 or less divides $x^{p^2}-x$ is going to be very helpful here. – eyeballfrog Jun 18 '21 at 15:39
  • @eyeballfrog Thanks! Does the following logic make sense for the first implication? Suppose $gcd(f(x), x^{p^2}-x) \neq 1$. Then, there exists $c \neq 1$ dividing both polynomials. This means $c$ is either $f(x)$ or $1$ both of which lead to a contradiction. How does the linked question relate to the other implication? – Nick Jun 19 '21 at 14:12
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    I think I understand. For the other implication, no irreducible polynomial of degree 2 or less can divide $f(x)$. So if some $c(x)$ divides $f(x)$, then it must have degree 3, 4, or 5. If $c(x)$ has degree 5, it is equal to $f(x)$. If $c(x)$ has degree 4, $f(x)$ factors into a product of two polynomials, one with degree 4 and the other with degree 1 which is a contradiction. The logic is similar in the case where $c(x)$ has degree 3. – Nick Jun 19 '21 at 14:45

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