If you restrict yourself to finite expressions, yes, that is true. However, if you allow such things as infinite series and the like, you can solve quintic equations---in terms of hypergeometric functions. See, eg.,It seems that Abel's theorem says that the quintic cannot be solved by arithmetic & root operations, but couldn't there be the situation where another function is used in concert with these operations?
Through Galois theory, it is proven that the general solution of a polynomial of degree at least 5 is not expressible in terms of the operations addition, multiplication (and their inverses) and taking roots.It seems that Abel's theorem says that the quintic cannot be solved by arithmetic & root operations, but couldn't there be the situation where another function is used in concert with these operations?