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Of course, we know the $A_5$ of $60$ order is an unsolvable group. But as the wiki here, there are also $12$ solvable groups in the same $60$ order still: enter image description here

Then I have generated many many irreducible polynomials whose Galois group is $60$ order with maple program to check, but none of them is solvable. What's wrong?

Is there a solvable polynomial whose Galois group is $60$ order? If it exists, can you please give any example?

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  • $\begingroup$ It doesn't work with degree $5$, because $S_5$ has no transitive subgroup of order $60$ except for $A_5$. But it works for degree $15$, see the table here. For example the solvable group $C_3⋊F_5$, where $F_5$ is the Frobenius group of order $20$. $\endgroup$ Commented May 20, 2022 at 12:39

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The irreducible polynomial $$ x^{15}-30x^{10}-3708x^5-2 $$ has Galois group $$ C_3\rtimes F_5, $$ which is solvable of order $60$. Here $F_5$ is the Frobenius group of order $20$.

Reference: The tables here.

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  • $\begingroup$ How did you obtain the polynomial by backpropagation from the Galois group? Does such an algorithm exist? $\endgroup$
    – mayi
    Commented May 20, 2022 at 12:50
  • $\begingroup$ No, in general reverse engineering is very difficult. In fact, the Inverse Galois Problem is still open in general. But here you can try to obtain an action of the Galois group on a suitable field, and then find the minimal polynomial of a primitive element. $\endgroup$ Commented May 20, 2022 at 13:01
  • $\begingroup$ In fact to find the solvable group of order 60 I have a simpler method than looking up the table in Mathematica. I just don't know how to construct the polynomial corresponding to it by these groups. As you said, this could be quite difficult. But can we be sure at the moment that polynomials of these groups are necessarily all exist? $\endgroup$
    – mayi
    Commented May 21, 2022 at 2:11
  • $\begingroup$ Yes, the transitive groups of order $60$ are just known. In this case, irreducible polynomials are also known, see the status of the Inverse Galois Problem, what is known and what is unknown. $\endgroup$ Commented May 21, 2022 at 8:30

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