Let $f(x)$ be a polynomial of degree $n$ over $\mathbb{Q}$, with Galois group isomorphic to the symmetric group $S_n$. How do I show that $f$ cannot be expressed as a composition $g(h(x))$ of two polynomials $g$ and $h$ of degrees > 1.


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    $\begingroup$ A Google search on "composition polynomials galois" returned the article ccms.or.kr/data/pdfpaper/jcms22_3/22_3_497.pdf as one of the top hits (did you try an internet search before asking your question?) and the paragraph after Lemma 3.2 will be of interest to you. $\endgroup$
    – KCd
    Apr 19 '15 at 3:40

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