2 ms·

Don't let the truth get in the way of a good story! hahahaha But yeah you're right edit: i don't recall Abel's proof, but Galois reformulation of what it mean

• arjvik 2y ago
  For a layman (I stopped short of Galois theory so far), what’s different about the permutation groups of quintic roots and above that leads to this?
  • will-burner 2y ago
    In lay terms the best I can say is that for n greater than or equal to 5, the set of all possible permutations of n things is complicated. For n less than 5 the set of all possible permutations of n things is simple just because n is small. That's what leads to there being general formulas for n = 2, 3, 4. Galois translated whether a polynomial has a solution for x in terms of the coefficients using algebraic operations up to using radicals into a property of the group of permutations of the roots of the polynomial. The property of the group is whether the group is solvable. For n greater than or equal to 5, the general permutation group on n objects is not solvable but for n less than 5 is is. There just are not that many permutation groups for n = 2, 3, and 4 objects and all these permutation groups are solvable. Generically a group is not solvable and so we see this with larger n.