We reprint an old math SE question here (see https://math.stackexchange.com/questions/1241224/composition-of-polynomials-and-galois-theory):

" 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 $> 1$. "

This old question does not have an answer, but one comment refers to the article

http://www.ccms.or.kr/data/pdfpaper/jcms22_3/22_3_497.pdf

of Choi. Therein, in the paragraph after Lemma 3.2, it is written:

"One of the important results about Gal$(f(g(x))/K)$ is that the Galois group is a wreath product of certain groups ([6])."

Here, $K$ denotes any field and the reference $[6]$ points to the article

"The Galois theory of iterates and composites of polynomials" by Odoni.

Alas, Choi does not give a particular Lemma or Theorem of $[6]$ as a reference.

The closest we could find is Lemma $4.1$ in $[6]$:

$K$ is an arbitrary field.

LEMMA $4.1.$ Let $f(g(X))$ be separable over $K$, and let deg$(f)= k$, deg$(g)=l$, with $k,l\geq 1$. Then $f(X)$ is also separable over $K$. Let $\mathcal{F}$ be Gal $f(X)/K$, identified with a subgroup of the permutations of its zeros in the usual way. Then there is an

injectivehomomorphism of Gal $f(g(X))/K$ into the wreath product of $\mathcal{F}$ with the symmetric group $S_l$.

The question is now:

How to derive the statement in Choi's article from Lemma $4.1$ of Odoni's article?

Or is there another result of Odoni's article needed?

Any additional references are very welcome.

EDIT:

I am interested in the question, if the Galois group of two such polynomials is a wreath product in a non-trivial way.

Thank you very much for the help.