Some fifth order equations have solutions that may be expressed as radicals. An example of such an equation is x^5=2, for which the fifth root of 2 as a solutio

Some fifth order equations have solutions that may be expressed as radicals. An example of such an equation is x^5=2, for which the fifth root of 2 as a solution. We know from school that there is a solution for any equation of degree 1 (linear) with integer coefficients using plain arithmetic, and degree 2 using radicals and arithmetic. The linked article mentions that the same holds for degrees 3 and 4. What Abel proved for degree 5 and Galois for any degree >= 5, is that for some equations of these degrees there's no expression involving radicals and arithmetic (*) that is a solution. To reiterate, Abel and Galois' results are about the very existence of a specific form of a solution, not "findability". (*) More technically: any finite formula involving composition of radicals, arithmetic operations and natural numbers.