Formulas for the roots of polynomials of the 5th degree

Every college student knows that the roots of a quadratic equation can be explicitly expressed by the coefficients of the equation. Moreover, only the four basic arithmetic operations together with the operation of extracting the square root are involved in the formula.

Similar formulas exist for polynomials of the 3rd and the 4th degree. These are more complex than in the case of the quadratic equation, but nevertheless the roots are again expressed by the coefficients of the equation.

It has been known for nearly 200 years (Ruffini, Abel, Galois) that no such explicit formulas expressing the roots of polynomials by their coefficients exist for polynomials of degree greater than 4, as long as the operations involved are limited to (a finite application of) the four basic operations and root extraction of arbitrary degree. This non-existence statement should not frighten the engineer who may fall upon an equation of the 5th degree. There are excellent techniques, like Newton's method of tangents, to compute an approximation of the polynomial roots with any prescribed accuracy.

Those who have followed a course in mathematical analysis may know that the primitive function of exponential of -x^2 cannot be expressed by a finite application of arithmetic operations and composition to elementary functions, without transition to the limit. In fact, this theorem is quite similar to that of Ruffini, Abel and Galois; the proofs of both use the same methods.