Problem 7 - Abstract Algebra: Theory and Applications

Question

Prove that the cosine of one degree \(\left(\cos 1^{\circ}\right)\) is algebraic over \(Q\) but not constructible.

Step-by-Step Solution

Answer

If yes, is it constructible? Answer: The cosine of 1 degree is algebraic over Q, but it is not constructible.

Step 1: Complex Numbers and De Moivre's theorem

First, we have to recall some facts about complex numbers. The cosine of an angle can be connected with complex numbers through Euler's formula: $$e^{i\theta} = \cos \theta + i \sin \theta$$ We are interested in $$\cos 1^{\circ} = \operatorname{Re}(e^{i\pi/180})$$ De Moivre's theorem connects powers of complex numbers with trigonometry: $$\left(e^{i\theta}\right)^n = e^{in\theta} = \cos n\theta + i \sin n\theta$$

Step 2: Applying De Moivre's theorem to \(\cos 1^{\circ}\)

Applying De Moivre's theorem to \(1^{\circ}\) (\(\pi/180\) radians) we get: $$\left(e^{i\pi/180}\right)^{360} = e^{i2\pi} =\cos 360^{\circ} + i \sin 360^{\circ} = 1$$ Since \(\cos^2 \theta + \sin^2 \theta = 1\): $$(\cos 1^{\circ} + i \sin 1^{\circ})^{360} = 1$$ Expanding the expression using the binomial theorem, we see that the real part of the expression (which is a polynomial of \(\cos 1^{\circ}\)) must be equal to 1, because all imaginary parts will cancel out, and the imaginary part of the expression (which is a polynomial of \(\sin 1^{\circ}\)) must be equal to 0. Thus, \(\cos 1^{\circ}\) is a root of a polynomial with rational coefficients, which makes it algebraic over \(Q\).

Step 3: Showing \(\cos 1^{\circ}\) is not constructible

To show that \(\cos 1^{\circ}\) is not constructible, we need to recall the result that an angle is constructible if and only if it is a multiple of a constructible angle that can be obtained using compass and straightedge constructions. The only constructible angles are multiples of \(3^{\circ}\) (by the cosine triple angle formula \(\cos 3 \theta = 4\cos^3\theta - 3\cos\theta\)), as well as their half, quarter and subsequent divisions by powers of 2, which are possible by bisection. Thus, neither a multiple nor a divisor of 1° by a power of 2 will not be a constructible angle, because they would never result in a \(3°\) multiple, showing that \(\cos 1^{\circ}\) is not constructible.

Key Concepts

De Moivre's theorem

De Moivre's theorem is a crucial bridge connecting complex numbers and trigonometry. It is particularly useful in raising complex numbers to powers. The theorem states that for any complex number in the form \(e^{i\theta}\), and any real number \(n\), the equation \(\left(e^{i\theta}\right)^n = e^{in\theta} = \cos(n\theta) + i\sin(n\theta)\) holds.

This expression allows you to determine the powers of complex numbers using trigonometric functions like cosine and sine. In practical terms, De Moivre's theorem is often applied when dealing with roots of unity or geometric rotations.

For example, in the problem concerning \(\cos 1^\circ\), applying this theorem helped evaluate \(\left(e^{i\pi/180}\right)^{360}\) to equal 1. Essentially, this uses the periodicity of sine and cosine, where \(\cos(360^\circ)\) and \(\sin(360^\circ)\) equal to zero, leading to unity. It's a powerful tool for manipulating and simplifying expressions involving complex numbers.

Constructible angles

Constructible angles are those which can be achieved using only a compass and straightedge. This fascinating geometric concept dates back to ancient Greek mathematicians who explored what could be created with simple tools.

For an angle to be constructible, it must be a rational multiple of \(3^\circ\), or must be possible to make from established constructible angles by repeatedly bisecting angles, which is dividing the angle by 2. Therefore, angles like \(60^\circ\), \(45^\circ\) and those resulting from their simple combinations or bisections fall into this category.

• An angle must adhere to these rules, a periodicity relating to powers of 2 to be constructible.
• Angles like \(1^\circ\) don't conform to the multiples or divisions satisfying these rules

This inability to bisect \(1^\circ\) down to a multiple of \(3^\circ\) makes \(\cos 1^\circ\) not constructible, though it remains algebraic over \(Q\).

Euler's formula

Euler's formula is an elegant equation that creates a profound link between exponential functions and trigonometric ones. Represented as \(e^{i\theta} = \cos \theta + i \sin \theta\), this formula highlights the beauty of mathematics by integrating elements like complex numbers \(i\), exponentials, and trigonometry.

Euler's formula provides the framework for understanding phenomena such as rotating frames or electrical engineering applications. It's foundational in fields like quantum mechanics and signal processing. What makes it stand out is its ability to express trigonometric functions as parts of complex exponentials, simplifying many calculations and proofs.

In practice, when attempting to understand trigonometric expressions or manipulate polynomials inclusive of sine and cosine, Euler's formula is indispensable. This is especially true for converting angles into complex forms, as seen in the exercise featuring \(\cos 1^\circ\). Such transformations make Euler's formula not just a theoretical marvel, but a practical tool for a wide array of mathematical applications.