Problem 4 - A Book of Abstract Algebra

Question

Let \(a(x)\) and \(b(x)\) be polynomials of positive degree. By the division algorithm, we may divide \(a(x)\) by \(b(x)\) $$ a(x)=b(x) q_{1}(x)+r_{1}(x) $$ Find the gcd of \(x^{3}+x^{2}+x+1\) and \(x^{4}+x^{3}+2 x^{2}+2 x\) in \(\mathbb{Z}_{3}[x]\).

Step-by-Step Solution

Answer

The gcd of \(x^3+x^2+x+1\) and \(x^4+x^3+2x^2+2x\) in \(\mathbb{Z}_3[x]\) is 1.

Step 1: Polynomial Division Setup

To find the greatest common divisor (gcd) of the polynomials \(a(x) = x^{4}+x^{3}+2x^{2}+2x\) and \(b(x) = x^{3}+x^{2}+x+1\) in \(\mathbb{Z}_{3}[x]\), we first need to set up the division. In \(\mathbb{Z}_{3}\), the coefficients are reduced modulo 3. Thus, division involves regular polynomial division where all calculations are reduced modulo 3.

Step 2: Perform Polynomial Division

Divide the leading term of \(a(x)\) by the leading term of \(b(x)\), which gives \(q_1(x) = x\). Compute \(q_1(x) \cdot b(x)\) and then subtract this product from \(a(x)\). Carry out the operations in \(\mathbb{Z}_{3}\).

Step 3: First Remainder Calculation

After performing the division, we obtain \(r_1(x) = a(x) - q_1(x) \cdot b(x)\). Simplifying, this gives us the remainder \(r_1(x) = x^2 + x + 2\).

Step 4: Continue with Euclidean Algorithm

Now, we continue using the Euclidean algorithm by dividing \(b(x) = x^3 + x^2 + x + 1\) by \(r_1(x) = x^2 + x + 2\). Perform the division similarly: divide the leading terms to find \(q_2(x)\) and subtract \(q_2(x) \cdot r_1(x)\) from \(b(x)\). This process is repeated until the remainder is zero.

Step 5: Find Next Remainder

After dividing, we obtain the next remainder \(r_2(x) = 2x + 1\). Now, repeat the division using \(r_1(x)\) and \(r_2(x)\). Perform polynomial division as before to find the next remainder.

Step 6: Final Remainder

Continue the division of \(r_1(x)\) by \(r_2(x)\): divide the leading terms, find \(q_3(x)\), perform multiplication and subtraction. The new remainder is \(r_3(x) = 1\), indicating that there is no further division possible.

Step 7: Identify the GCD

The last non-zero remainder before reaching 1 is the gcd of the polynomials. In our case, since \(r_3(x) = 1\), the gcd is the constant polynomial \(1\).

Key Concepts

Greatest Common Divisor

The Greatest Common Divisor (GCD) is a vital concept in mathematics, particularly in polynomial arithmetic. For polynomials, the GCD is the largest polynomial that divides two given polynomials without leaving a remainder. It's similar to finding the GCD of numbers, but instead, we work with expressions like polynomials.

This concept is crucial when simplifying polynomial expressions or solving polynomial equations, as it helps to reduce the complexity of the problem. In the context of the given exercise, our task is to find the GCD of the polynomials \(x^3+x^2+x+1\) and \(x^4+x^3+2x^2+2x\) in the field \(\mathbb{Z}_3[x]\), which involves working with coefficients reduced modulo 3.

Finding the GCD helps us understand the underlying structure of polynomials by revealing common divisors and simplifying equations.

Division Algorithm

The Division Algorithm is a fundamental concept not just for numbers but also for polynomials. It states that for any two polynomials \(a(x)\) and \(b(x)\) (where \(b(x)\) is not zero), it is possible to express \(a(x)\) as:

• \(a(x) = b(x)q(x) + r(x)\)

This means that \(a(x)\) can be divided by \(b(x)\) to produce a quotient \(q(x)\) and a remainder \(r(x)\), where the degree of \(r(x)\) is less than the degree of \(b(x)\).

In polynomial terms, the division algorithm allows us to repeatedly divide the polynomials, reducing them systematically. This process is central to finding the GCD since we continue dividing until the remainder is zero. At this point, the last non-zero remainder is the GCD. This repetitive division and reduction process is the core of the exercise at hand.

Euclidean Algorithm

The Euclidean Algorithm is a beautiful and efficient method for finding the GCD of two polynomials. Starting with two polynomials, we apply successive divisions using the division algorithm. Each step reduces the degree of the polynomials, and we repeat the process until the remainder is zero.

Steps involved in the Euclidean Algorithm:

• Divide the larger polynomial by the smaller one.
• Take the remainder and divide it by the preceding remainder.
• Continue this process until a remainder of zero is achieved.
• The last non-zero remainder is the GCD.

This step-by-step process is what we apply in the given exercise. It simplifies the task of finding the GCD in a structured and systematic way.

Modulo Arithmetic

Modulo Arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value, known as the modulus. In this exercise, the modulus is 3, as indicated by \(\mathbb{Z}_3[x]\). This means all coefficients of the polynomials are reduced modulo 3.

When performing operations like addition, subtraction, multiplication, or division of coefficients, we always take the result modulo 3. For example, if the sum of coefficients is 5, it is reduced to 2 because 5 modulo 3 equals 2.

This technique helps manage polynomial divisions effectively within a finite set of values, ensuring the operations remain within the limits of modulo 3. It's a key part of simplifying and solving polynomial expressions under this exercise.

Polynomial Remainder

The remainder in polynomial division signifies what is left after dividing one polynomial by another. For polynomials \(a(x)\) and \(b(x)\), if the remainder is \(r(x)\), then the degree of \(r(x)\) is always less than \(b(x)\).

In our exercise, each division step results in a remainder. These remainders play a crucial role as they are used in subsequent divisions when applying the Euclidean Algorithm.

Steps involving polynomial remainders:

• Compute the remainder when dividing two polynomials.
• Use this remainder for the next division with the previous divisor.
• Continue until zero is reached, indicating no further division is possible.

Understanding how to compute and utilize remainders is critical for this problem-solving method, as they lead us to the GCD.