General polynomial of degree $n$ is irreducible from Gauss' Lemma

Question

I'm studying "old-fashioned" Galois theory and the following is an elementary and fundamental problem to keep proceed with my studies. I'm really stuck at that question.

Can someone help me please?

If $s_1, ..., s_n$ are independent variables over the complex field, then the general polynomial of degree $n$: $$f(x) = x^n - s_1x^{n-1} + \cdots + (-1)^n s_n$$ is irreducible in $\mathbb{C} (s_1, ..., s_n) [x]$.

My professor gave the following tip:

Use the following consequence of Gauss' Lemma: Let $R$ be a Unique Factorization Domain, $F$ its fraction field and $p(x) \in R[x]$ a non-constant primitive (i.e., the gcd of its coefficients is 1) polynomial. Then $p(x)$ is irreducible in $F[x]$ if and only if is irreducible in $R[x]$.

I think we also can use Girard's relations because of the format of the polynomial, I don't know.

Thank you

Answer 1

Your teacher suggested to consider $f$ as element of $\mathbb{C} [s_1, ..., s_n] [x]=\mathbb{C} [s_1, ..., s_n, x]$. Thus $f$ becomes a polynomial in $n+1$ variables. It's obvious now that $f$ is irreducible since its degree as a polynomial in $s_n$ is one.