Question

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

Short answer

\( \omega = x \bmod (x^n-1) \) is not a primitive nth root of unity in \( R \) due to lack of lower order other than identity.

Step-by-step solution

Understand the Representation in R

In the given context, \( R = F[x] / \langle x^n - 1 \rangle \) means that we are considering the factor ring of the polynomial ring \( F[x] \) over the ideal generated by \( x^n - 1 \). The element \( \omega = x \mod (x^n - 1) \) means we are looking at the equivalence class of \( x \) in this quotient ring.

Define a Primitive nth Root of Unity

A primitive \( n \)th root of unity is an element \( \alpha \) such that \( \alpha^n = 1 \) and for no smaller positive integer \( m \), \( \alpha^m = 1 \) unless \( m = n \). The order of \( \alpha \) is then \( n \).

Check the Order of \( \omega \)

Evaluate \( \omega^n \) in the ring \( R \). We find that \( \omega^n = x^n \equiv 1 \mod (x^n - 1) \) since \( x^n - 1 \equiv 0 \) in \( R \). This satisfies the condition necessary for being an \( n \)th root of unity.

Determine If \( \omega \) Is Primitive

Since \( \omega = x \) satisfies \( \omega^n = 1 \) in \( R \), we must check if \( \omega \) has order strictly less than \( n \). Compute \( \omega^k = x^k \equiv 1 \mod (x^n - 1) \) for \( k < n \). As there is no such \( k \) since the polynomial \( x^n - 1 \) has no roots in \( R \) other than those derived from its factorization over \( F \), it implies that the minimal polynomial satisfies higher than needed conditions for unit roots.

Conclude that \( \omega \) Is Not Primitive

Although \( \omega^n = 1 \), the equation \( x^k = 1 \mod (x^n - 1) \) for a smaller unique \( k \) does not exist, meaning \( \omega \) cannot achieve primitivity within \( R \) as the division is made irrelevant by identity unification constraints.

Key concepts

Polynomial Rings

Polynomial rings are fundamental in algebra and are similar to the more familiar integer rings. A polynomial ring, denoted as \( F[x] \), is the set of all polynomials where the coefficients are drawn from a field \( F \). This setup forms a ring structure because it allows addition, subtraction, and multiplication of polynomials. A key feature is that division is not always possible unless specific conditions are met, like those involving the degree of the polynomials.

• Each polynomial can be expressed in the form \( a_0 + a_1x + a_2x^2 + \, \cdots \, + a_nx^n \), where \( a_i \) are elements of the field \( F \).
• Polynomials are added or multiplied by applying rules that are similar to numerical addition and multiplication, except they involve handling powers of \( x \).
• The degree of a polynomial is the highest power of \( x \) with a non-zero coefficient.

Understanding polynomial rings is crucial for studying algebraic structures, especially when dealing with more complex entities like factor rings.

Factor Rings

Factor rings, also known as quotient rings, are created by dividing a ring by one of its ideals. In the context of polynomial rings, when we consider a polynomial ring \( F[x] \) and an ideal \( \langle g(x) \rangle \), the factor ring \( F[x] / \langle g(x) \rangle \) consists of all equivalence classes of polynomials modulo \( g(x) \).

• An ideal \( \langle g(x) \rangle \) is generated by a single polynomial \( g(x) \), which contains all multiples of \( g(x) \).
• In the factor ring, two polynomials are considered equivalent if their difference is divisible by \( g(x) \).
• Operations in factor rings follow these equivalence relations, simplifying many polynomial problems to a more manageable form.

This structure is beneficial in simplifying polynomials without losing important properties, and it lays much groundwork in abstract algebra for solving polynomial equations.

Primitive Roots

Primitive roots are essential in understanding polynomial roots, especially cyclic roots of unity. A primitive \( n \)th root of unity is a complex number \( \alpha \) that satisfies two main conditions:

• \( \alpha^n = 1 \), meaning that \( \alpha \) raised to the power \( n \) returns 1, a root of unity.
• \( \alpha^m eq 1 \) for any positive integer \( m < n \), ensuring that \( \alpha \) is not a simpler root of unity.

Primitive roots are critical in fields like number theory and digital signal processing as they ensure maximum cycle orders and fully span the necessary group of roots for \( n \).Not every root that satisfies the unity condition is primitive, as some may have lower orders, implying an intricate hierarchy among potential solutions. Recognizing these differences is vital when working with cyclic equations, particularly in algebraic contexts like fields and rings.