Math Newsletter number 17; Wednesday, November 17, 2010.
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Finding the five fifth roots of 1.

It is a consequence of the fundamental theorem of algebra
that the equation x**5 - 1 = 0, has 5 roots.

There are 5 numbers x such that x**5 = 1.

Can we find these 5 numbers exactly?

One of them is trivial.

1**5 = 1.

Since 1 is a fifth root of 1,
x**5 - 1 factors as (x-1)*some 4th degree polynomial.

In fact, we can find exactly that 4th degree polynomial by
formally dividing (x-1) into (x**5 - 1)

If we do, we get the result, that
(x**5 -1)/(x-1) = x**4 + x**3 + x**2 + x + 1.

This result is a special case of the general pattern,

(x-1)(x+1) = x**2 -1
(x-1)(x**2 + x + 1) = x**3 - 1
(x-1)(x**3 + x**2 + x + 1) = x**4 - 1
(x-1)(x**4 + x**3 + x**2 + x + 1) = x**5 - 1
etc

So, to find the other 4 fifth roots of 1, we need only
factor x**4 + x**3 + x**2 + x + 1

A good guess might be that x**4 + x**3 + x**2 + x + 1 factors
as (x**2 + g x + 1)(x**2 + h x + 1).

One reason that this is a good guess is tied into the fact
that if x is a fifth root of 1, then so is (1/x).

Multiply out our quadratics to get:

(x**2 + g x + 1)(x**2 + h x + 1) = x**4 + x**3 + x**2 + x + 1

x**4 +(g+h)x**3 + (gh+2)x**2 + (g+h)x + 1
=x**4 + x**3 + x**2 + x + 1

Immediately we see, by matching coefficients, that

g+h = 1
gh +2 = 1

or

g+h = 1
gh = -1

Solving first of these two equations for g gives,
g=(1-h).

Plugging into second of these two equations gives,

(1-h)h = -1

Multiply through by -1

(h-1)h = 1
h**2 - h = 1

h**2 - h - 1 = 0

Solving this quadratic equation for h yields,

h = (1+sqrt(5))/2

Incidently, This number, (1+sqrt(5))/2, is famous. It is
called the "Golden Ratio".

g = 1 - h = 1 - (1+sqrt(5))/2 = 2/2 - (1+sqrt(5))/2

g = (1 - sqrt(5))/2

(x**2 + [(1 - sqrt(5))/2] x + 1)(x**2 + [(1+sqrt(5))/2]x + 1)
= x**4 + x**3 + x**2 + x + 1.

You may confirm that this is correct by direct calculation.

Now it has become easy to find the other fifth roots.

Two of the other fifth roots are the solutions to

(x**2 + [(1 - sqrt(5))/2] x + 1) = 0

and the other two fifth roots are the solutions to

(x**2 + [(1+sqrt(5))/2]x + 1) = 0.

Thus the four other fifth roots of 1 are:

root 1
=[(-1 + sqrt(5))/2]/2 + sqrt( [(1+sqrt(5))/2]**2 - 4)/2
=[(-1 + sqrt(5))/4] + sqrt([(6+2sqrt(5))/4] - 4)/2
=[(-1 + sqrt(5))/4] + sqrt([(1/4)([6+2sqrt(5)] - 4*4)/2
=[(-1 + sqrt(5))/4] + (1/2)sqrt(6+2sqrt(5)-16)/2
=[(-1 + sqrt(5))/4] + [sqrt(6+2 sqrt(5) - 16]/4
=[(-1 + sqrt(5))/4] + [sqrt(-10 + 2 sqrt(5))]/4
=([-1 + sqrt(5)] + sqrt(-1) sqrt[10 - 2 sqrt(5)] )/4

The other roots are

([-1 + sqrt(5)] - sqrt(-1) sqrt[10 - 2 sqrt(5)] )/4

([-1 - sqrt(5)] + sqrt(-1) sqrt[10 + 2 sqrt(5)] )/4

([-1 - sqrt(5)] - sqrt(-1) sqrt[10 + 2 sqrt(5)] )/4