Question

Let \(f \in R[x]\) for a Unique Factorization Domain \(R\). Show that \(f=\mathrm{pp}(f)\) if and only if \(f\) is primitive.

Short answer

\(f = \mathrm{pp}(f)\) if and only if \(f\) is primitive.

Step-by-step solution

Understanding Primitive Polynomials

A polynomial \(f\) is called primitive if the greatest common divisor of its coefficients is 1. Thus, \(f\) is primitive if there is no non-unit element in \(R\) that divides all coefficients of \(f\).

Definition of Primitive Part

The primitive part of a polynomial \(f\), denoted \(\mathrm{pp}(f)\), is obtained by dividing \(f\) by the gcd of its coefficients. This results in a primitive polynomial.

Proving the Forward Implication

Assume \(f = \mathrm{pp}(f)\). By definition of \(\mathrm{pp}(f)\), \(f\) must already be primitive, as \(\mathrm{pp}(f)\) is primitive. Thus, the forward implication is proven: if \(f = \mathrm{pp}(f)\), then \(f\) is primitive.

Proving the Reverse Implication

Assume \(f\) is primitive. According to the definition, the gcd of its coefficients is 1. Thus, \(\mathrm{pp}(f)\) which divides \(f\) by this gcd gives \(f\) itself. Hence, \(\mathrm{pp}(f) = f\), proving the reverse implication: if \(f\) is primitive, then \(f = \mathrm{pp}(f)\).

Key concepts

Primitive polynomials

In algebra, understanding primitive polynomials is essential, especially when studying polynomials over unique factorization domains (UFDs). A polynomial is considered "primitive" if there is no non-unit element in the ring that can divide each of its coefficients. This means that the greatest common divisor (GCD) of all its coefficients is 1. For example, the polynomial \(f(x) = 2x^2 + 3x + 1\) over the integers is primitive because there isn't an integer greater than 1 that divides all its coefficients (2, 3, and 1).
In simpler terms, if you can find any number larger than 1 that fits evenly into all terms, then the polynomial isn't primitive.
Being a primitive polynomial is a fundamental property because it helps identify when a polynomial cannot be further simplified or factored over the same ring other than by units. This characteristic is useful when doing polynomial division, ensuring polynomials are simplified to their core forms for further calculations.
When working with polynomials, always check for primitiveness as a first step. It's like ensuring a fraction is in its simplest form before doing any further math with it!

Greatest common divisor

The greatest common divisor (GCD) is a concept many might recognize from basic arithmetic, but it also plays a crucial role in higher-level math, particularly with polynomials. For two or more numbers, the GCD is the largest positive integer that divides each of the numbers without a remainder. The concept is similar when applied to polynomials.
When we talk about the GCD of polynomial coefficients, like in our earlier example \(2x^2 + 3x + 1\), we look for the largest polynomial degree or integer that can evenly divide all coefficients. If this GCD is 1, the polynomial is deemed primitive.
Understanding and finding the GCD of coefficients is essential for simplifying polynomials and establishing their primitive nature. It's similar to reducing a fraction to its lowest terms to facilitate easier calculations and better understand its properties in equations.

• For polynomials, checking the GCD of coefficients helps us determine if the polynomial can be simplified further or if it is already in its simplest form.
• This concept is crucial when proving properties about polynomials or when performing operations like polynomial division.

Recognizing the GCD's role can provide clarity and allow for a deeper comprehension of polynomial structures.

Primitive part of a polynomial

The primitive part of a polynomial, often abbreviated as \(\mathrm{pp}(f)\), serves as an excellent tool to understand and manipulate polynomials. It's like simplifying a fraction down to its lowest terms. To get to the "primitive part," you divide the polynomial by the greatest common divisor (GCD) of its coefficients.
Imagine you have the polynomial \(4x^2 + 8x + 12\). The coefficients are 4, 8, and 12, and the GCD is 4. Dividing each term by 4, you get the primitive polynomial \(x^2 + 2x + 3\), which is the primitive part.
This concept is important because the primitive part of a polynomial helps in simplifying complex algebraic expressions and ensuring that you are working with the simplest form of the polynomial. It also assists in understanding whether a polynomial is primitive. If \(f = \mathrm{pp}(f)\), then the polynomial is primitive.

• Finding the primitive part is a key step in preparing a polynomial for more complex operations, such as factoring or integration.
• It allows mathematicians and students to quickly judge the primitiveness of polynomials and ensure that calculations involving them will be as straightforward as possible.

The primitive part of a polynomial not only aids in simplification but also ensures efficiency in broader mathematical computations.