### The impossibility of solving general quintics in radicals

The fundamental theorem of algebra states that every polynomial equation of degree `n` has `n` roots in the complex plane (counting multiplicity). The picture on the left shows `log[1 + |q[z]|]` where `q` is a quintic. The picture on the right shows the lines where the real and imaginary parts of `q[z]` are zero; they cross at right angles at the roots.

The general solution of the quadratic equation was found more than 4000 years ago. The solutions of the cubic and quartic--found in the 1500s--were major results of Renaissance mathematics.

Mathematicians struggled for centuries to find formulas for the solutions of equations of higher degree, but despite the efforts of Euler, Bezout, Malfatti, Lagrange, and others, no general solutions were found. Finally, Ruffini (1799) and Abel (1826) showed that the solution of the general quintic cannot be written as a finite formula involving only the four arithmetic operations and the extraction of roots. By 1832 Galois had developed the theory of Galois groups and described exactly when a polynomial equation is solvable.

The colored squares represent the Galois groups of the equation:

in the `a-b` plane for integers `a` and `b`. Gray squares represent quintics that factor into lower degree polynomials. Orange and red squares represent those quintics that don't factor but that can still be solved in radicals. The dark and light blue squares correspond to quintics that cannot be solved in radicals.

There are polynomials of degree greater than four which do not factor over the rationals, but that can still be solved in radicals. An example is the quintic:

one of whose roots is

### Tschirnhaus's transformation

In 1683 Tschirnhaus published Method of Eliminating All Intermediate Terms from a Given Equation. Although the title is exaggerated, Tschirnhaus's paper was the most important idea for the solution of algebraic equations in about 200 years.

Tschirnhaus's transformation reduces the `n`th degree polynomial equation:

to one with up to three fewer terms:

by transforming the roots as follows:

where the `gamma`j can be expressed in radicals in terms of the `a`j. Thus every quintic can be transformed into one of the form:

The `b`j can ultimately be expressed in radicals in terms of the `a`j. The resulting expressions are typically enormous; for a general quintic with symbolic coefficients they require megabytes of storage.

### Klein's approach to the quintic

The group structure of the icosahedral equation is related to the stereographic projection of the triangulated regular icosahedron.

In 1877 Klein published Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree. In this remarkable book and later article Klein gave a complete solution of the quintic.

He used a Tschirnhaus transformation to reduce the general quintic to the form:

He found the solution of this reduced quintic by first solving the related icosahedral equation:

where `Z` can be expressed in radicals in terms of `a`, `b`, and `c`.

A solution of the icosahedral equation using hypergeometric functions is:

### Solutions based on series

Johann Lambert (1757) seems to have been the first person to have the idea of using series to solve a polynomial equation. In the next 150 years similar ideas were raised independently by Leonard Euler (1770), Pafnuti Chebyshev (1838), and Gotthold Eisenstein (1844), among others.

We first illustrate the idea with the quadratic and then go on to the quintic. Consider the generic quadratic polynomial:

Let `r[y]` be a branch of the inverse function of `q` in a small neighborhood about `y = 0`, meaning that `q[r[y]] = y` for any sufficiently small `y`. Then `r[0]` is clearly a root of `q`.

In order to find `r[0]` as a funtion of `a`, `b`, and `c`, we will compute the formal power series of `r[y]` in `y` at zero; for this it suffices to invert `q[x]` as a formal power series:

In[1]:=

Out[1]=

This gives nine terms of the series, which we evaluate at zero to get `r[0]`:

In[2]:=

Out[2]=

The general term is:

In[3]:=

We can sum this series to get a closed-form solution for one of the roots of the original quadratic equation:

In[4]:=

Out[4]=

Now we show how this method works for a quintic of the form:

Proceeding as above, we compute:

In[5]:=

Out[5]=

This gives the first eight terms in the series for one of the roots. The general term of this series can be written as:

In[6]:=

This series can be summed using hypergeometric functions; one of the roots of the quintic is:

In[7]:=

Out[7]=

In principle this method will work for any polynomial.

Galois groups of quintics are related to the symmetries of the icosahedron.

### Solutions based on differential equations

Cockle (1860) and Harley (1862) developed a method for solving algebraic equations based on differential equations.

We illustrate their approach with the quintic equation:

In[1]:=

Each root is a function of the paremeter `rho`. The differential resolvent of this quintic is a linear differential equation of order four of the form:

In[2]:=

The coefficients `a`j are polynomial functions in `rho`. To find them explicitly, we differentiate the quintic equation with respect to `rho` and substitute these derivatives into the differential equation. This gives a `4`th degree algebraic equation in `t[rho]`. Setting the coefficients of the powers of `t[rho]` to zero gives a system of five linear equations in the `a`j. Solving this system gives the differential resolvent:

In[3]:=

We solve the differential equation `Omega`:

In[4]:=

This is a general solution to the differential equation that depends on the four parameters `c1`, `c2`, `c3`, and `c4`. We will find the values of these parameters so that this solution will also satisfy the original quintic. To do this we could substitute this general solution into the quintic and solve the resulting equation.

However, it is enough to substitute the series expansion of the solution at `rho = 0`, where the main term of the series expansion of the generalized hypergeometric function is `1`:

In[5]:=

In[6]:=

Now we collect like terms in `rho` and set the coefficients equal to zero. We solve the resulting system of four linear equations for `c`j:

In[7]:=

In[8]:=

Finally we have obtained the five roots of the given quintic:

The approach just described can be used for polynomials of any degree. However, only polynomials that can be written as trinomials of the form:

have roots that can be represented in terms of hypergeometric functions of one variable.

### Equations of higher degree

Some of the ideas described here can be generalized to equations of higher degree. The basic ideas for solving the sextic using Klein's approach to the quintic were worked out around 1900. For algebraic equations beyond the sextic, the roots can be expressed in terms of hypergeometric functions in several variables or in terms of Siegel modular functions.