**Theorem.** The Galois group of a quintic polynomial $f\in\mathbb{Q}[x]$ is $A_5$ if and only if its discriminant is a rational square and its Weber sextic resolvent has no rational root.

**Question.** What are known infinite families of of quintic polynomials in $\mathbb{Q}[x]$, each with Galois group $A_5$?

I am aware of two explicit such families as follows.

**Example 1.** $f(x)=x^5+(5t^2-1)(5x+4)$ for $t\in\mathbb{Z}$ such that $t\equiv\pm 1\pmod{21}$. This is Exercise 3.7.2 in the book *Generic Polynomials* by Jensen, Ledet and Yui.

**Example 2.** $f(x)=x^5+(t^2-5^5)(x-4)$ for non-zero $t\in\mathbb{Q}$ such that $\forall u\in\mathbb{Q}: t\ne g(u)$ where $g(u)=\frac{(u^3-18u^2+8u-16)(u^3+2u^2+18u+4)}{2u^2(u^2+4)}$. This is from the paper *Reducibility and the Galois group of a parametric family of quintic polynomials* by Lavallee, Spearman and Williams.

**Note.** Both are all trinomials, presumably because it is easier to construct polynomials with square discriminant when some coefficients vanish. It would be nice to see otherwise.