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 $A_n$.

## 1 Answer

Hermez and Salinier, Rational trinomials with the alternating group as Galois group, Journal of Number Theory, Volume 90, Issue 1, September 2001, Pages 113-129 has the abstract,

For any integer $n\ge7$, we show how to explicitly build an infinite number of rational trinomals of degree $n$ whose Galois group over $\bf Q$ is isomorphic to $A_n$.

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