Problem 15 - Algebra: Pure and Applied

Question

Let \(G\) be the multiplicative group of all \(n\) th roots of unity. If \(a \in G,\) what is \(a^{1} ?\)

Step-by-Step Solution

Answer

In group G, for any element \(a\), \(a^1 = a\).

Step 1: Understanding the nth Root of Unity

The multiplicative group of all nth roots of unity consists of all complex numbers of the form \(e^{2\pi ik/n}\) for \(k = 0, 1, 2, \ldots, n-1\). These numbers represent the solutions to the equation \(x^n = 1\) in the complex plane.

Step 2: Recognizing the Element of G

Each element \(a\) in the group \(G\) can be described by \(a = e^{2\pi ik/n}\) with some integer \(k\) and satisfies \(a^n = 1\).

Step 3: Evaluating Exponents in Group

Since we need to find \(a^1\), we simply raise the element \(a = e^{2\pi ik/n}\) to the power of 1. This does not change the element, so \(a^1 = a\).

Key Concepts

Multiplicative Group

A multiplicative group is a fascinating concept in mathematics, specifically in algebra. It is a group whose operation is multiplication. For the nth roots of unity, the group consists of complex numbers that, when multiplied a certain number of times (n times, in this case), give 1. These numbers form what is known as the multiplicative group of nth roots of unity.
These groups have specific properties that make them quite interesting:

• Closure: If you multiply any two nth roots of unity, you'll get another nth root of unity.
• Associativity: This property always holds true, as it does in the multiplication of numbers.
• Identity: There is an element in the group that, when multiplied by any element, leaves the element unchanged (here, it is 1).
• Inverses: For every element in the group, there exists another element that results in the multiplicative identity when multiplied together.

These traits mean every operation within the group remains within the group, making it closed under multiplication.

Complex Numbers

Complex numbers are a set of numbers that expand our number system by combining real and imaginary numbers. An imaginary number is defined as the square root of a negative number, with the basic imaginary unit being represented as 'i', where \(i^2 = -1\).
A complex number is expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(bi\) represents the imaginary part. These numbers can be graphed on a complex plane, where the x-axis signifies the real part and the y-axis signifies the imaginary part.
In terms of nth roots of unity, complex numbers play a crucial role. The nth roots are specific types of complex numbers that, when raised to the power of n, result in a real number, 1. These roots have geometric interpretations where they are spread evenly on a circle in the complex plane, each spread at an angle equivalent to \(360/n\) degrees.
This geometric representation aids with the understanding and applications of complex numbers, such as in various fields like physics and engineering.

Exponents in Group

Exponents are widely used across mathematics to express repeated multiplication. In a group setting like the multiplicative group of nth roots of unity, exponents help us understand the structure and properties of the group.
When considering elements of the group, represented as \(e^{2\pi ik/n}\), raising an element \(a\) to an exponent \(m\) can be seen as performing the group operation (multiplication, in this case) \(m\) times. This forms a strong link between group theory and exponential functions.
To solve our original problem where we need to know what \(a^1\) is within the group, we utilize the properties of exponents in groups that state \(a^1 = a\). When an element is raised to the power of 1, its value remains unchanged. This gives a clear understanding that within the group, applying the exponent does not alter the base element, hence simplifying computations in group operations involving powers.