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

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.

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 $$\Bbb F_p[X]$$.