Question

Let \(R\) be a ring (commutative, with 1 ), \(f, g \in R[x]\), and \(n \in \mathbb{N}\). Prove that \(f \equiv g \bmod x^{n+1}\) implies \(f^{\prime} \equiv g^{\prime} \bmod x^{n}\), and give an example where \(f^{\prime} \equiv g^{\prime} \bmod x^{n+1}\) does not hold.

Short answer

If \( f \equiv g \bmod x^{n+1} \), then \( f' \equiv g' \bmod x^n \). Example: \( f(x) = x^{n+1}, g(x) = 0 \); \( f' \not\equiv g' \bmod x^{n+1} \).

Step-by-step solution

Understanding the problem

We need to prove that if two polynomials \( f \) and \( g \) in the ring \( R[x] \) are congruent modulo \( x^{n+1} \), then their derivatives \( f' \) and \( g' \) are congruent modulo \( x^n \). This uses properties of polynomial differentiation and congruence.

Expressing congruence condition

Since \( f \equiv g \bmod x^{n+1} \), it means there exists a polynomial \( h(x) \in R[x] \) such that \( f(x) = g(x) + x^{n+1}h(x) \). This is the key assumption we will use to show the relationship between the derivatives.

Differentiating both sides of the equation

Differentiate both sides of the equation: \( f(x) = g(x) + x^{n+1}h(x) \). The derivative of \( f(x) \) is \( f'(x) \), and the derivative of \( g(x) + x^{n+1}h(x) \) is \( g'(x) + (n+1)x^nh(x) + x^{n+1}h'(x) \).

Using the congruence relation for derivatives

Since \((n+1)x^n h(x) + x^{n+1} h'(x)\) is divisible by \(x^n\) (it equals \((n+1)x^n h(x)\) plus a term divisible by \(x^{n+1}\)), we have \( f'(x) - g'(x) = (n+1)x^n h(x) + x^{n+1} h'(x) \). Thus, \( f' \equiv g' \bmod x^n \).

Example where stronger congruence fails

Consider \( f(x) = x^{n+1} \) and \( g(x) = 0 \) in \( R[x] \). Then, \( f' = (n+1)x^n \) and \( g' = 0 \). Thus, \( f' \equiv 0 \bmod x^n \) but not \( \bmod x^{n+1} \), since \( (n+1)x^n ot\equiv 0 \bmod x^{n+1} \) unless \( (n+1) = 0 \) in \( R \).

Key concepts

Ring Theory

Ring theory is a branch of abstract algebra that explores sets equipped with two binary operations: addition and multiplication. In a ring, these operations must satisfy certain conditions such as associativity, and the presence of a multiplicative identity (often denoted as "1"). Rings can be either commutative or non-commutative.

• Commutative Ring: A ring where the multiplication operation is commutative, i.e., for any two elements \( a \) and \( b \) in the ring, \( a \times b = b \times a \).
• With Unity: A ring that contains a multiplicative identity element, meaning there exists an element "1" such that for any element \( a \) in the ring, \( a \times 1 = a \).

Rings are crucial in many areas of mathematics, especially when dealing with polynomial congruences as they provide the framework within which congruences like \( f \equiv g \bmod x^{n+1} \) are considered. Understanding ring properties helps in drawing conclusions about polynomial behavior under modular conditions.

Polynomial Differentiation

Polynomial differentiation is a calculus concept applied to polynomials, where the derivative of a polynomial provides a new function that represents its rate of change.

• Basic Idea: If \( f(x) \) is a polynomial in the ring \( R[x] \), its derivative \( f'(x) \) is calculated by finding the derivative of each term.
• Rule for Power Functions: For a monomial \( ax^k \), the derivative is \( akx^{k-1} \).

In the context of polynomial congruence, differentiation helps in analyzing how two related polynomials (in terms of congruence) behave similarly or differently. Differentiating both sides of a congruence often sheds light on a deeper congruence relationship at a different level, such as proving \( f' \equiv g' \bmod x^n \) from \( f \equiv g \bmod x^{n+1} \). The derivative of the polynomial expressive nature can introduce additional congruences as observed with terms like \( (n+1)x^n h(x) \).

Modular Arithmetic

Modular arithmetic is the mathematics of congruences, where numbers "wrap around" upon reaching a certain value known as modulus. In simpler terms, it is like arithmetic of a clock.

• Congruence Relation: Two numbers \( a \) and \( b \) are congruent modulo \( n \) if they leave the same remainder when divided by \( n \), written as \( a \equiv b \bmod n \).
• Application to Polynomials: Polynomial congruences extend the idea of congruences to polynomial expressions, allowing us to explore equivalencies of entire polynomial functions under specific moduli.

In the context of ring theory, we often use modular arithmetic to show how certain properties or operations remain consistent, even when simplified under a particular modulus. For example, if \( f(x) \equiv g(x) \bmod x^{n+1} \), it implies that the difference \( f(x) - g(x) \) is divisible by \( x^{n+1} \), which then can have implications for their derivatives, proving statements like \( f' \equiv g' \bmod x^n \). Exploring non-trivial examples solidifies our understanding of the limits of congruence like the failure example provided in the solution.