Question

Let \(D\) be an infinite integral domain, and let \(g, h \in D[X] .\) Show that if \(g(x)=h(x)\) for all \(x \in D,\) then \(g=h .\) Thus, for an infinite integral domain \(D,\) there is a one-to-one correspondence between polynomials over \(D\) and polynomial functions on \(D\).

Short answer

Question: Show that there is a one-to-one correspondence between polynomials over an infinite integral domain \(D\) and polynomial functions on \(D\). Answer: If two polynomials \(g(x)\) and \(h(x)\) in an infinite integral domain \(D\) are equal for all \(x \in D\), then the polynomials must be the same, ensuring a one-to-one correspondence between polynomials and polynomial functions on \(D\).

Step-by-step solution

1. Define a difference polynomial

First, let's define another polynomial \(f(x) = g(x) - h(x)\), which represents the difference between the given polynomials \(g(x)\) and \(h(x)\). Since the polynomials \(g(x)\) and \(h(x)\) have the same values for all \(x \in D\), this implies that \(f(x) = 0\) for all \(x \in D\).

2. Identify a non-zero coefficient

We are given that \(D\) is an infinite integral domain, therefore it has an infinite number of elements. Now, let's assume \(g(x)\) and \(h(x)\) are different polynomials, which means \(f(x)\) is a non-zero polynomial. This means that \(f(x)\) must have at least one nonzero coefficient, let's say \(a_k\), with the corresponding degree \(k \geq 0\).

3. Consider the roots of the difference polynomial

Since \(f(x)\) is a non-zero polynomial, it has a degree, and let's say \(k\) is the smallest degree among all nonzero coefficients of \(f(x)\). Now, we are given that \(f(x) = 0\) for all \(x \in D\). This implies that the polynomial \(f(x)\) has an infinite number of roots, which are all the elements of \(D\).

4. Derive a contradiction using the integral domain properties

Using the property of integral domains where nonzero elements have no zero divisors, we can derive a contradiction here. Since \(D\) is an infinite integral domain, having an infinite number of roots in \(f(x)\) means there must be another polynomial \(p(x)\) such that \(f(x) = x^k p(x)\). However, we chose the coefficient \(a_k\) with the smallest degree among all nonzero coefficients in \(f(x)\), which means there's no such polynomial \(p(x)\) that can satisfy this condition.

5. Conclude that the polynomials are equal

Since we have derived a contradiction by assuming that \(g(x)\) and \(h(x)\) are different polynomials, our assumption must be false. Therefore, we conclude that \(g(x) = h(x)\) for all \(x \in D\). This shows that there is a one-to-one correspondence between polynomials over \(D\) and polynomial functions on \(D\) for an infinite integral domain.

Key concepts

Polynomials

A polynomial is a mathematical expression that consists of variables, coefficients, and non-negative integer exponents. In simple terms, think of polynomials as sums of algebraic terms like monomials or combinations of these terms.
For example, the polynomial \( p(x) = 3x^2 + 2x + 1 \) is made up of three terms: \( 3x^2 \), \( 2x \), and \( 1 \).

Polynomials are a foundational element in mathematics, especially in algebra and calculus. They have several key characteristics:

• Coefficients: These are the numbers in front of the variables, such as 3 and 2 in our example.
• Degree: This is the highest power of the variable in the polynomial, which is 2 in \( p(x) = 3x^2 + 2x + 1 \).
• Variables: Represented by symbols such as \( x \), \( y \), etc., upon which operations are performed.

Understanding these components makes it easier to grasp more complex concepts involving polynomials.

Infinite Domain

An infinite domain in mathematics refers to a set with an infinite number of elements.
This concept is crucial when discussing infinite integral domains because it implies that within such a domain, there are untold possibilities for values that a variable can take.

In this context, consider an infinite integral domain \( D \), which is a set equipped with both addition and multiplication operations that adhere to the rules of arithmetic like no zero divisors, meaning if \( ab = 0 \) then either \( a = 0 \) or \( b = 0 \).

• Integral Domain: Includes entities like the integers. It prevents division by zero and ensures unique factorization.
• Applications: Infinite domains are often considered when analyzing polynomial behaviors because a polynomial that is zero on an infinite domain means it must be identically zero, implying identical functions over that domain.

This infinite nature assists in proving fundamental properties between polynomials in such domains.

Polynomial Functions

Polynomial functions are functions that can be written in the form of a polynomial.
They play a vital role in various branches of mathematics due to their simple form yet powerful applications.

Every polynomial can be viewed as a polynomial function, but what's critical here is the relationship between polynomial expressions and their functions over specific domains.

• Function Definition: A polynomial function \( f(x) \) might look like \( a_nx^n + a_{n-1}x^{n-1} + \,\cdots\, + a_1x + a_0 \), where the \( a_i \) are coefficients and \( n \) marks the highest degree.
• Characteristic: In an infinite domain, if two polynomial functions agree on all values in that domain, the polynomials are identical. This stems from how polynomials uniquely determine their functions based on the values they take.
• Versatility: These functions are used in curve fitting, computer graphics, and real-world modeling due to their adaptability and ease of calculation.

Understanding polynomial functions means witnessing the bridge between abstract algebraic operations and tangible mathematical modeling.