Question

Use Gauss's method to solve the cyclotomic equation \(x^{6}+\) \(x^{5}+x^{4}+x^{3}+x^{2}+x+1=0\)

Short answer

Answer: The four distinct solutions are \(1\), \(-1\), \(i\), and \(-i\).

Step-by-step solution

Identify the roots of the equation in terms of complex roots of unity

Gauss's method involves using the complex roots of unity, \(e^{2 \pi i k/n}\), where \(n\) is the order of the root and \(k = 0, 1, ..., n-1\). The equation given is a cyclotomic equation of degree 6, as it's a sum of all powers of \(x\) up to 6, plus 1. So, let's consider a 6th root of unity, \(z = e^{2 \pi i k/6}\).

Check the roots of the equation

Now let's see if \(z\) satisfies the given equation. The sum of all powers of \(z\) up to 6 and 1 is: $$z^6 + z^5 + z^4 + z^3 + z^2 + z + 1$$ We need to verify if this sum is equal to zero.

Find the geometric series sum

The expression given above is a geometric series with a common ratio of \(z\), so its sum can be computed as follows: $$S = \frac{1 - z^7}{1 - z}$$ As we are dealing with a sixth-order root of unity, \(z^6 = 1\) and therefore, \(z^7 = z\). Replacing this in the expression, we get: $$S = \frac{1 - z}{1 - z}$$ So, the sum is equal to 1 for all \(z \neq 1\). Therefore, none of the sixth roots of unity satisfy this equation.

Consider 12th roots of unity

Since the 6th roots of unity didn't work, let's consider the 12th roots of unity. Let's have \(w = e^{2 \pi i k/12}\). In this case: $$w^{12} = 1$$ $$\Rightarrow w^6 = \pm 1$$ The sum of all powers of \(w\) up to 6 and 1 is: $$w^6 + w^5 + w^4 + w^3 + w^2 + w + 1$$ The geometric series sum can be computed as before: $$S = \frac{1 - w^7}{1 - w}$$

Simplify the expression for the sum

Before, we substituted \(w^6 = 1\). But now we substitute \(w^6 = -1\), which gives us: $$S = \frac{1 + w}{1 - w}$$ We need this sum to be zero. For it to be zero, we need \(1 + w = 0\). Therefore, \(w = -1\). So, the twelve roots of unity that satisfy the cyclotomic equation are the ones for which \(w = e^{2 \pi i k/12} = -1\).

Determine the appropriate 12th roots of unity

Since \(w = -1\), we can rewrite it as \(w = e^{i \pi}\), which implies \(k = 6\) (because \(-1\) can be represented as a 12th root of unity when \(k = 6\)). So, the 12th root of unity that solves the cyclotomic equation is \(e^{i \pi/2}\).

Express the solutions of the cyclotomic equation

Now, we will express the solutions of the cyclotomic equation using the 12th root of unity \(x = w^k\). The solutions are: $$x_1 = e^{\pi i/2}$$ $$x_2 = e^{\pi i}$$ $$x_3 = e^{3 \pi i/2}$$ $$x_4 = e^{2 \pi i}$$ $$x_5 = e^{5 \pi i/2}$$ $$x_6 = e^{3 \pi i}$$

Simplify the expressions for the solutions

Finally, we can simplify the expressions for the solutions of the cyclotomic equation: $$x_1 = i$$ $$x_2 = -1$$ $$x_3 = -i$$ $$x_4 = 1$$ $$x_5 = i$$ $$x_6 = -1$$ Since \(x_1 = x_5\) and \(x_2 = x_6\), we have four distinct solutions: \(1, -1, i, -i\).

Key concepts

Gauss's Method

Gauss's method helps in solving cyclotomic equations by utilizing the roots of unity, which are complex answers to polynomial equations equalling zero. This method takes advantage of symmetries in these equations that relate to highly structured mathematical forms called cyclotomic polynomials.

• These polynomials look like a geometric series with a ratio of the root of unity.
• By identifying the order of the cyclotomic polynomial, the method examines the specific forms of the roots that would satisfy the equation.
• Gauss's method efficiently narrows down potential solutions by recognizing that certain roots don't satisfy the polynomial, focusing only on the roots that do.

In our case, when addressing the equation \(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 = 0\), Gauss’s method begins by examining the 6th roots of unity. This is because the equation itself suggests its association with these roots. By identifying these roots and analyzing their behavior, we determine which satisfy the equation under their respective scenarios.

Roots of Unity

Roots of unity are essentially the solutions to the equation \(x^n = 1\), meaning they are specific complex numbers that together form a unit circle on the complex plane. These roots can be expressed as \(e^{2\pi i k/n}\), where \(n\) is the total number of roots and \(k\) is a whole number ranging from 0 to \(n-1\).

• The 6th roots of unity, specifically, represent six equidistant points on the unit circle grid.
• These points help in simplifying complex polynomials, revealing information about their structure and possible solutions.

In our cyclotomic equation, these roots were initially tested, but it was discovered they did not satisfy the equation since their combination summed to 1, not 0. We then moved on to consider 12th roots of unity, recalibrating the focus effectively on these until reaching solutions.

Geometric Series

A geometric series is a sum of terms where each term is a constant multiple of the previous one. In our exercise, the equation resembles such a series with the common ratio being a root of unity.

• This configuration helps simplify finding the sum of the polynomial provided.
• For example, the series formed is \(z^6 + z^5 + z^4 + z^3 + z^2 + z + 1\).

For an \(n\)th root of unity like \(z\), the series sum formula becomes \(S = \frac{1 - z^{n+1}}{1 - z}\). Upon testing with both 6th and 12th roots of unity, the series helps determine whether each set of roots truly sums to zero, a crucial check in finding valid solutions.

Complex Numbers

Complex numbers come into play significantly when dealing with cyclotomic equations and tools like Gauss's method. They are expressed as \(a + bi\), where \(i\) is the imaginary unit, satisfying \(i^2 = -1\). These numbers are essential because they embody the concept of rotating around the complex plane, a key geometric characteristic of roots of unity.

• The exponential form \(e^{i\theta}\) of complex numbers is particularly useful in handling roots of unity.
• It uses Euler's formula, \(e^{i\theta} = \cos(\theta) + i\sin(\theta)\), to express complex numbers as points on the unit circle.
• Understanding how to maneuver these numbers allows us to find solutions like \(i, -1, -i,\) and \(1\) for our cyclotomic equation.

In our exercise, the 12th roots of unity were evaluated and provided the solutions in complex form, revealing both real and purely imaginary parts as resolutions.