In a previous question on the same topic I asked about explicit examples of quintic polynomials having integer coefficients with Galois Group $A_5$.
Answers and comments established the following:
(i) Some examples
(ii) A theorem of Dedekind which shows how factorisations modulo $p$ tell you cycle types which are definitely present in the Galois group
(iii) It is possible to achieve $A_5$ in a polynomial with five real roots, hence we can't assume that complex conjugation is the only source of a transposition (this latter observation is more obvious than it first seems, but conjugation is the easy route to an element of order 2 in the Galois Group)
(iv) One can exclude a transposition (hence verify that only even permutations occur) if the discriminant is a square
My question is mainly about point (iv). It is a mechanical process to compute the discriminant of a quintic - so in one sense this is already "easy". However it is not an easy process by hand or by memory - unless someone persuades me otherwise. And my main question is whether there is an example where this can be avoided through some other observation or trick, which makes the example "easier" to remember.