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