It is my understanding that the polynomial $f_n(x)=x^n-1$ has the Galois Group of $C_n$, cyclic group of order $n$. In some sense, these are the "simplest" polynomials with that Galois Group. Is there a formula for the polynomial, say $g_n(x)$, whose Galois group is $A_n$? And, I mean $g_n(x)$ in the same sense as $f_n(x)$: that is, the "simplest" polynomials with that Galois Group of $A_n$.

It is my understanding that the polynomial $f_n(x)=x^n-1$ has the Galois Group $(\mathbb{Z}/n\mathbb{Z})^*$, group of units of order $\phi(n)$. In some sense, these are the "simplest" polynomials with that Galois Group. Is there a formula for the polynomial, say $g_n(x)$, whose Galois group is $A_n$? And, I mean $g_n(x)$ in the same sense as $f_n(x)$: that is, the "simplest" polynomials with that Galois Group of $A_n$.