Problem 22 - A First Course in Abstract Algebra

Question

. Find a polynomial of degree \(>0\) in \(Z_{4}[x]\) that is a unit.

Step-by-Step Solution

Answer

The polynomial \( x + 3 \) is a unit in \( \mathbb{Z}_4[x] \).

Step 1: Understanding the Problem

We aim to find a polynomial with coefficients in the ring of integers modulo 4, \( \mathbb{Z}_4[x] \), such that the polynomial is invertible in this ring. This means there should exist another polynomial that, when multiplied with our given polynomial, yields the polynomial 1 (the multiplicative identity).

Step 2: Choosing a Polynomial

Consider the constant polynomial \( f(x) = 1 \) in \( \mathbb{Z}_4[x] \). Although it is of degree 0, it is a trivial unit since any constant non-zero element of \( \mathbb{Z}_4 \) is a unit. However, we need a degree greater than 0.

Step 3: Constructing a Polynomial of Degree 1

A common choice is \( f(x) = x + 3 \) in \( \mathbb{Z}_4[x] \). Here, \( f(x) = x + 3 \) has a degree of 1 and both coefficients are elements of \( \mathbb{Z}_4 \).

Step 4: Verifying Invertibility

To verify \( x+3 \) is a unit, we need to find another polynomial \( g(x) \) such that \( (x + 3)g(x) \equiv 1 \mod 4 \). Since the polynomial is linear, try \( g(x) = 3x + 1 \).

Step 5: Multiplying the Polynomials

Calculate \( (x + 3)(3x + 1) = x \cdot 3x + x \cdot 1 + 3 \cdot 3x + 3 \cdot 1 \). Simplifying, this yields \( 3x^2 + x + 9x + 3 \equiv 0 + 4x + 3 \equiv 1 \mod 4 \).

Step 6: Conclusion

Thus, \( (x + 3)(3x + 1) \equiv 1 \mod 4 \), confirming that \( x + 3 \) is a unit in \( \mathbb{Z}_4[x] \).

Key Concepts

Invertible Polynomial

An invertible polynomial, also known as a 'unit polynomial,' is a polynomial within a given ring that has a multiplicative inverse. This means if you have a polynomial \( f(x) \), there exists another polynomial \( g(x) \) such that \( f(x) \cdot g(x) = 1 \), where 1 is the polynomial equivalent of the multiplicative identity in the ring. In simple terms, when an invertible polynomial is multiplied by its inverse, it results in the identity element.

• Resetting the identity means reaching an outcome that behaves neutrally in addition, such as 1 for multiplication.
• Not every polynomial is invertible. To verify invertibility involves some trials or systematic checking in the context of the specific ring.

The concept of invertible polynomials is essential when working with polynomial rings as it allows for operations such as division to be possible within the scope of the given algebraic structure.

Ring of Integers Modulo

The ring of integers modulo a number, often denoted as \( \mathbb{Z}_n \), consists of integers from 0 to \( n-1 \), where any operations are performed modulo \( n \). This creates a finite set of residues resulting from division by \( n \). A characteristic feature of this ring is that:

• Addition and multiplication are defined such that results "wrap around" upon reaching \( n \).
• A prime modulus \( n \) simplifies the structure, often making it easier to find inverses of integers.

In the context of polynomial rings such as \( \mathbb{Z}_4[x] \), both the polynomial coefficients and resulting operations adhere to modulo \( 4 \). This constraint heavily influences which polynomials can be classified as invertible or units in the ring.

Polynomial Degree

The degree of a polynomial is a fundamental but simple concept defined by the highest power of \( x \) with a non-zero coefficient. For example, in the polynomial \( x^2 + 3x + 1 \), the degree is 2 because of the \( x^2 \) term.

• Understanding polynomial degree is key because it informs us of the polynomial's potential behaviors and characteristics.
• Degree constraints often play a central role in exercises involving finding certain types of polynomials.

In our problem, we are asked to find polynomials of degree greater than zero, suggesting that more complexity is recognized compared to constant polynomials.

Polynomial Multiplication

Polynomial multiplication is a straightforward extension of basic arithmetic multiplication applied to polynomials. When multiplying two polynomials, you systematically apply the distributive property, multiplying each term in one polynomial by every term in the other polynomial. Here’s a quick step-by-step on how to do it:

• Every term in the first polynomial is multiplied by every term in the second polynomial.
• The products are added together, and like terms are combined.

For example, multiplying \( (x + 3) \) by \( (3x + 1) \), each term in \( (x + 3) \) is combined with each in \( (3x + 1) \), then simplified, results in \( x \cdot 3x + x \cdot 1 + 3 \cdot 3x + 3 \cdot 1 \). Understanding this operation is pivotal in confirming the invertibility of polynomials within rings like \( \mathbb{Z}_4[x] \).