Let \(\mathbb{F}_p\) be the finite field with \(p\) elements, and let \(f(x) \in \mathbb{F}_p[x]\) be a polynomial of degree 4.
Show that \(f(x)\) is irreducible if and only if the polynomials \(f(x)\) and \(x^{p^2} - x\) are relatively prime.
Assuming \(f(x)\) is irreducible over \(\mathbb{F}_p\), describe the Galois group of \(f(x)\) over \(\mathbb{F}_p\). Include a generating set for the Galois group in your answer.
Solution
1. The monic irreducible factors of \(x^{p^2}-x\) over \(\FF_p\) are exactly the monic irreducibles of degree \(1\) and \(2\).
In a field \(\FF_{p^2}\) with \(p^2\) elements, every element satisfies \(\alpha^{p^2}=\alpha\), so \(x^{p^2}-x=\prod_{\alpha\in\FF_{p^2}}(x-\alpha)\) has distinct roots. The minimal polynomial over \(\FF_p\) of \(\alpha\in\FF_{p^2}\) has degree \([\FF_p(\alpha):\FF_p]\), which divides \(2\). Conversely, a monic irreducible \(g\) of degree \(d\in\{1,2\}\) has a root \(\beta\) with \(\FF_p(\beta)\) of order \(p^d\), so \(\beta^{p^d}=\beta\) and hence \(\beta^{p^2}=\beta\); thus \(g\), the minimal polynomial of \(\beta\), divides \(x^{p^2}-x\).
2. In part (a), \(f\) is irreducible if and only if \(\gcd(f,x^{p^2}-x)=1\).
If \(f\) is reducible, it has a monic irreducible factor of degree \(1\) or \(2\), since \(\deg f=4\); by step 1 that factor divides \(x^{p^2}-x\), so the gcd is not \(1\). If \(f\) is irreducible, a common monic irreducible factor with \(x^{p^2}-x\) would be \(f\) itself, of degree \(4\), contradicting step 1; so the gcd is \(1\).
3. In part (b), the Galois group of an irreducible \(f\) over \(\FF_p\) is cyclic of order \(4\), generated by the Frobenius automorphism \(\sigma(x)=x^p\).
Adjoining one root \(\alpha\) of \(f\) gives \(\FF_p(\alpha)\) of degree \(4\), a field \(\FF_{p^4}\) with \(p^4\) elements. The extension \(\FF_{p^4}/\FF_p\) is Galois with cyclic group of order \(4\) generated by \(\sigma\) [@DF04], so it contains every root of the irreducible polynomial \(f\) and is its splitting field.
4. Q.E.D.
Steps 2 and 3 answer parts (a) and (b).