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    An irreducible in $\FF_p[x]$ divides $x^{p^n}-x$ iff its degree divides $n$

    P-VTQYH

    Area
    Algebra
    Topics
    Finite Fields, Irreducibility Criteria, Polynomials
    Status
    draft

    Let \(\pi(x)\in\FF_p[x]\) be irreducible of degree \(d\). Prove \[ \pi(x)\mid x^{p^n}-x \iff d\mid n. \]

    Solution

    Let \(\alpha\) be a root of \(\pi\) in an algebraic closure of \(\FF_p\). Since \(\pi\) is irreducible of degree \(d\), \[ \FF_p(\alpha)\cong\FF_{p^d}. \]

    Now \[ \pi(x)\mid x^{p^n}-x \] if and only if \(\alpha\) is a root of \(x^{p^n}-x\), i.e. \[ \alpha^{p^n}=\alpha. \] The roots of \(x^{p^n}-x\) are exactly the elements of \(\FF_{p^n}\). Hence this is equivalent to \[ \FF_p(\alpha)=\FF_{p^d}\subseteq\FF_{p^n}. \]

    Finite-field subfields satisfy \[ \FF_{p^d}\subseteq\FF_{p^n} \iff d\mid n. \] Therefore \[ \pi(x)\mid x^{p^n}-x \iff d\mid n. \]

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