Question

Show that \(w=x \bmod \left(x^{n}-1\right) \in R=F|x| /\left\langle x^{n}-1\right\rangle\), where \(F\) is a field of churacteristic not dividing \(n\), is not a primitive \(n\)th root of unity for \(n \geq 2\).

Short answer

Since \( w^n = 0 \neq 1 \) in \( R \), \( w \) is not a primitive \( n \)th root of unity.

Step-by-step solution

Understand the Problem

We need to show that the element \( w = x \mod (x^n-1) \) in the ring \( R = F|x| / \langle x^n-1 \rangle \) is not a primitive \( n \)th root of unity. This means \( w^n eq 1 \) in the ring \( R \).

Calculate \( w^n \) in the Ring \( R \)

In the ring \( R = F|x| / \langle x^n-1 \rangle \), elements are polynomials reduced modulo \( x^n-1 \). To find \( w^n \) where \( w = x \mod (x^n - 1) \), we calculate \( x^n \mod (x^n - 1) \). Since \( x^n \equiv 0 \mod (x^n - 1) \), this implies \( w^n = x^n = 0 \) in \( R \).

Conclude \( w \) is not a Primitive \( n \)th Root of Unity

A primitive \( n \)th root of unity \( \zeta \) must satisfy \( \zeta^n = 1 \) and \( \zeta^k eq 1 \) for any \( 1 \leq k < n \). Since we found \( w^n = 0 \) in \( R \), and not equal to 1, \( w \) cannot be a primitive \( n \)th root of unity.

Key concepts

Modular Arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after reaching a certain value—the modulus. In simple terms, it's like looking at the remainder when you divide one number by another. When we work with polynomials, particularly in polynomial rings, we use similar concepts.

Within the context of the given problem, you are working in a polynomial ring that's been reduced modulo another polynomial, specifically the polynomial \(x^n - 1\). This means you are dealing with polynomials but considering them equivalent if they differ by a multiple of \(x^n - 1\).

Thus, any polynomial \(p(x)\) is considered the same as another polynomial \(q(x)\) if \(p(x) - q(x)\) is divisible by \(x^n - 1\). This is similar to saying 10 is equivalent to 2 modulo 8 in simple integer modular arithmetic, as both have a remainder of 2 when divided by 8. This concept is crucial for simplifying calculations and understanding the behavior of elements in these rings.

Primitive Roots of Unity

In mathematics, a primitive root of unity is a special complex number that has a product value of 1 when multiplied by itself a certain number of times, specifically \(n\) times, but does not equal 1 when multiplied any fewer times before \(n\). These roots are fundamental in number theory and are particularly useful in various fields like signal processing and cryptography.

To better understand this concept, consider \(\zeta_n\) as a primitive \(n\)th root of unity. This means \(\zeta_n^n = 1\), but \(\zeta_n^k eq 1\) for any \(1 \leq k < n\). In our exercise, we are examining whether \(w = x \mod (x^n-1)\) can be such a root. The solution shows that \(w^n = 0\) in the ring \(R\), rather than 1, indicating that it cannot be a primitive root of unity.

This concept is pivotal when working with polynomials in modular arithmetic, especially when the modulus is of the form \(x^n - 1\), as it helps determine cyclic properties and symmetries within the constructed algebraic structures.

Fields and Characteristics

Fields are algebraic structures in which you can perform addition, subtraction, multiplication, and division without leaving the structure. Each field has a characteristic, which is a fundamental number that defines the field's behavior with respect to arithmetic operations.The characteristic of a field is the smallest number of times you need to add the multiplicative identity (1) to get zero. For example, for the field of rational numbers, the characteristic is zero because adding one any number of times will never yield zero.In the context of this problem, we are told the characteristic of field \(F\) doesn't divide \(n\). This implies that repeated addition won't simplify the problem in a trivial way. It ensures that certain cyclic properties related to roots of unity within polynomial rings behave in predictable and useful ways.

Understanding the characteristic is crucial for delving into more advanced topics in abstract algebra, such as understanding the behavior of polynomials over different fields.