Polynomials and Polynomial Functions in I_m = Z/mZ

Hence the set of all polynomial functions is a subset of the set of all functions. Therefore, if the set of all functions is finite, then the subset of polynomial functions must also be finite.
  • #1
Math Amateur
Gold Member
MHB
3,998
48
I am reading Joseph J.Rotman's book, A First Course in Abstract Algebra.

I am currently focused on Section 3. Polynomials

I need help with the a statement of Rotman's concerning the polynomial functions of a finite ring such as \(\displaystyle \mathbb{I}_m = \mathbb{Z}/ m \mathbb{Z} \)

The relevant section of Rotman's text reads as follows:https://www.physicsforums.com/attachments/4530In the above text we read the following:

" ... ... The reader should realize that polynomials and polynomial functions are distinct objects. For example, if \(\displaystyle R\) is a finite ring (e.g. \(\displaystyle \mathbb{I}_m\)), then there are only finitely many functions from \(\displaystyle R\) to itself; a fortiori, there are only finitely many polynomial functions. ... ... "

My question is as follows:

How, exactly (indeed, rigorously and formally) can we demonstrate that only finitely many functions from \(\displaystyle R\) to itself implies that there are only finitely many polynomial functions ... ... indeed, how would we prove this statement ...Hope someone can help ... ...

Peter
 
Physics news on Phys.org
  • #2
Peter said:
I am reading Joseph J.Rotman's book, A First Course in Abstract Algebra.

I am currently focused on Section 3. Polynomials

I need help with the a statement of Rotman's concerning the polynomial functions of a finite ring such as \(\displaystyle \mathbb{I}_m = \mathbb{Z}/ m \mathbb{Z} \)

The relevant section of Rotman's text reads as follows:In the above text we read the following:

" ... ... The reader should realize that polynomials and polynomial functions are distinct objects. For example, if \(\displaystyle R\) is a finite ring (e.g. \(\displaystyle \mathbb{I}_m\)), then there are only finitely many functions from \(\displaystyle R\) to itself; a fortiori, there are only finitely many polynomial functions. ... ... "

My question is as follows:

How, exactly (indeed, rigorously and formally) can we demonstrate that only finitely many functions from \(\displaystyle R\) to itself implies that there are only finitely many polynomial functions ... ... indeed, how would we prove this statement ...Hope someone can help ... ...

Peter

If a set i finite then each of its subsets is also finite. Here we have the set of all the functions mapping $R$ into $R$ finite if $R$ is finite. A polynomial function is, in particular, a function.
 

FAQ: Polynomials and Polynomial Functions in I_m = Z/mZ

1. What is a polynomial?

A polynomial is an algebraic expression consisting of variables, coefficients, and exponents. It can have multiple terms, each with a different degree. Examples of polynomials include 3x^2 + 5x + 2 and 2y^3 + 4y^2 + y + 7.

2. What is a polynomial function?

A polynomial function is a function that is defined by a polynomial. It takes a specific input value and returns a corresponding output value based on the polynomial equation. For example, the polynomial function f(x) = 2x^3 + 5x + 1 takes an input value of x and returns an output value based on the equation.

3. What is I_m = Z/mZ in relation to polynomials?

In mathematics, I_m = Z/mZ is the notation for the set of integers modulo m, also known as the integers modulo m or the integers modulo m ring. It is used in polynomial and polynomial function equations to represent the set of coefficients and variables.

4. How do you perform operations with polynomials?

To perform operations with polynomials, you can use the standard algebraic rules such as combining like terms, distributing, and factoring. Addition and subtraction of polynomials involve combining like terms, while multiplication and division involve distributing and factoring.

5. What is the degree of a polynomial?

The degree of a polynomial is the highest exponent of the variable in the expression. For example, the polynomial 3x^2 + 5x + 2 has a degree of 2, while the polynomial 4y^3 + 7y + 1 has a degree of 3. The degree of a polynomial determines its overall shape and behavior.

Back
Top