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Find a monic integer polynomial of degree $n$ whose Galois group over $\mathbb Q$ is $S_n$.

It's an exercise in Serge Lang's Algebra, and he suggests to show that there exists a polynomial whose reductions (mod some primes) are such that the Galois group contains enough cycles to generate $S_n$ and use Chinese remainder theorem.

My understanding:

If $n$ is prime then it has actually been shown by an example, it is:

$X^5-X-1\equiv(X^2+X+1)(X^3+X^2+1)\bmod\ 2$

and since its Galois group over $\mathbb Q$ contains a $5$-cycle and a $2$-cycle (deduced from the reduction form), then we get $S_n$.

So for n=prime we can construct $f(X)\equiv (X^2+X+1)X^{n-2}\bmod\ 2$ and $f(X)\equiv monic\ irreducible \ polynomial\ of\ degree\ n\bmod\ 3$. It's existed by Chinese remainder theorem and irreducible over $\mathbb Q$. Its Galois group contains a n-cycle (by n is prime ) and a 2-cycle (by constructed) and hence is $S_n$.

But for general case I can't construct a n-cycle.

Or if I have "every transitive subgroup of $S_n$ which contains $S_{n-1}$ as a subgroup is $S_n$" (I don't know if this is true) I can do it by induction.

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This is Theorem $26.4$ on page $112$ in these lecture notes. It uses, as you said, CRT and Dirichlet's theorem, starting from the fact that for each prime $p$ there exists an irreducible polynomial of degree $n$ for all $n$ in $\Bbb F_p[X]$.

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