Steps to the Quintic
It is easy to solve a quadratic equation in Mathematica:
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The cubic equation is also straightforward, but the answer is
more complicated. It is shown in a very small font size in order
to give an idea of how large it is without filling up your screen.
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The quartic can also be solved, but now the answer is quite long, so in
order to avoid choking your link, it is not displayed here.
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What about the quintic? Here is what happens if we try to solve
it directly.
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Root objects are an implicit way to represent the
solution. They can be differentiated and expanded out in series,
and with approximate numerical values for the coefficients,
they immediately yield a numerical solution. Of course, we can
solve a quintic with numerical coefficients immediately by using
the built-in Mathematica function NSolve.
Ruffini (1799) and Abel (1826) proved that it is not possible to
give an explicit solution for the general quintic equation with
symbolic coefficients in terms of square roots, cube roots, and
so on. Is there an explicit solution to the quintic with symbolic
coefficients? Yes! In the late 1800s, several mathematicians
constructed such solutions. However, it was necessary to go beyond
the extraction of roots and to use elliptic and hypergeometric
functions. Mathematica can handle these higher mathematical
functions in the same way as ordinary trigonometric or exponential
functions. Combined with Mathematica's algebraic capabilities,
this makes it possible to implement various symbolic solutions to
the quintic.
Here is a complete implementation of Hermite's solution of the
quintic in Mathematica in terms of elliptic functions. In
the general case it takes almost a trillion bytes to store the
explicit formulas in terms of the symbolic coefficients. The program
below provides a much shorter, yet complete, representation of the
solution.
Hermite's solution proceeds in three steps. In the first step one
eliminates the quartic and cubic terms; in the next step one eliminates
the square term; in the final step one solves a reduced equation of
the form:
PrincipalTransform[p == 0, x, y] eliminates the quartic
and cubic terms of the generic quintic p of the form:
The x is the variable in p, and the
y will be the variable of the new quintic. The output
is a pair whose first element is the transformation in the form of
a pure function and whose second element is the new quintic itself,
called the principal quintic.
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Psi[q, x, n] calculates the sum of the
nth powers of all roots of the quintic.
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BringJerrardTransform[p == 0, y, z] eliminates the
quadratic term of the principle quintic p of the
form:
5 2
b0 y + b3 y + b4 y + b5 = 0
The y is the variable of p, and the
z will be the variable of the new quintic. Again,
the output is a pair whose first element is the transformation in
the form of a pure function and whose second element is the new
quintic itself, this time called a Bring-Jerrard quintic.
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CanonicalTransform[p == 0. z, t] normalizes the
linear term of the Bring-Jerrard quintic poly to be -1.
The z is the variable of p, and the
t will be the variable of the new quintic. Again,
the output is a pair whose first element is the transformation in
form of a pure function and whose second element is the new quintic,
called the canonical quintic.
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HermiteQuinticSolve solves the canonical quintic.
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Start with the following quintic. To save space we use numerical
coefficients:
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The first Tschirnhaus transformation eliminates the quartic and
cubic terms:
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After the second Tschirnhaus transformation the quadratic term is
also gone:
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The next step makes the coefficient of the linear term equal to
-1:
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Now we calculate the solution of this quintic:
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To get the solutions of the original equation, we must reverse the
three transformations that brought the original equation to the reduced
one. The first step is a simple linear transformation which gives the
roots of the Bring-Jerrard quintic:
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Inverting the (nonlinear) Tschirnhaus transformations produces extraneous
reults that must be dropped:
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These are the solutions of the principal quintic:
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Finally we get the five roots of the original quintic:
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