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I have seen this lemma given without proof in some articles (see example here), and I guess it is well known, but I couldn't find an online reference for a proof.

It states like this:

Let $K$ be a field and $f,g \in K[x]$. Let $\alpha$ be a root of $f$ in the algebraic closure of $K$. Then $f \circ g$ is irreducible over $K$ if and only if $f$ is irreducible over $K$ and $g-\alpha$ is irreducible over $K(\alpha)$.

Can you please give a proof for this?

  • 1
    It may be in Schinzel's book about polynomials.2012-04-17
  • 1
    Check M.C.R. Butler, Reducibility Criteria for Polynomials of Two General Classes, PLMS, 1(1957), 63-74.2012-05-27
  • 0
    @Gerry Myerson Or maybe not!2012-07-28

1 Answers 1