Problem 1 - 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 \(b(x)\) $$ a(x)=b(x) q_{1}(x)+r_{1}(x) $$ Prove that every common divisor of \(a(x)\) and \(b(x)\) is a common divisor of \(b(x)\) and \(r_{1}(x)\). It follows from part 1 that the ged of \(a(x)\) and \(b(x)\) is the same as the ged of \(b(x)\) and \(r_{1}(x)\). This procedure can now be repeated on \(b(x)\) and \(r_{1}(x)\); divide \(b(x)\) by \(r_{1}(x)\) : Next, $$ \begin{aligned} r_{1}(x) &=r_{2}(x) q_{3}(x)+r_{3}(x) \\ & \vdots \end{aligned} $$ Finally, $$ r_{n-1}(x)=r_{n}(x) q_{n+1}(x)+0 $$ In other words, we continue to divide each remainder by the succeeding remainder. Since the remainders continually decrease in degree, there must ultimately be a zero remainder. But we have seen that $$ \operatorname{gcd}[a(x), b(x)]=\operatorname{gcd}\left[b(x), r_{1}(x)\right]=\cdots=\operatorname{gcd}\left[r_{n-1}(x), r_{n}(x)\right] $$ Since \(r_{n}(x)\) is a divisor of \(r_{n-1}(x)\), it must be the ged of \(r_{n}(x)\) and \(r_{n-1}(x)\). Thus, $$ r_{n}(x)=\operatorname{gcd}[a(x), b(x)] $$ This method is called the euclidean algorithm for finding the gcd.

Step-by-Step Solution

Answer

The Euclidean Algorithm finds the gcd by successive polynomial divisions, reducing degrees until reaching zero.

Step 1: Understand the Division Algorithm

The division algorithm allows us to divide a polynomial \(a(x)\) by another polynomial \(b(x)\) where \(a(x) = b(x) q_1(x) + r_1(x)\) and the remainder \(r_1(x)\) satisfies \(0 \leq \deg(r_1(x)) < \deg(b(x))\). This serves as the foundation for the Euclidean Algorithm used to find the gcd of two polynomials.

Step 2: Define Common Divisors

A common divisor of \(a(x)\) and \(b(x)\) is a polynomial that divides both \(a(x)\) and \(b(x)\) without leaving a remainder. We want to prove that any common divisor \(d(x)\) of \(a(x)\) and \(b(x)\) is also a common divisor of \(b(x)\) and \(r_1(x)\).

Step 3: Use the Division Equation

Since \(d(x)\) divides both \(a(x)\) and \(b(x)\), it must divide any linear combination of them by properties of divisors. From \(a(x) = b(x) q_1(x) + r_1(x)\), we can write \(r_1(x) = a(x) - b(x) q_1(x)\). Since \(d(x)\) divides both \(a(x)\) and \(b(x)\), it divides the combination \(a(x) - b(x)q_1(x)\), meaning \(d(x)\) divides \(r_1(x)\). Thus, \(d(x)\) is a common divisor of both \(b(x)\) and \(r_1(x)\).

Step 4: Apply Repeated Division

To find the gcd of \(a(x)\) and \(b(x)\), the process continues by dividing \(b(x)\) by \(r_1(x)\) to produce \(b(x) = r_1(x) q_2(x) + r_2(x)\), then \(r_1(x)\) by \(r_2(x)\), continuing this process until a zero remainder occurs.

Step 5: Conclude with the GCD

At each step, the new gcd is equal to the gcd of the dividend and remainder from the previous step. Since the degree of remainders decreases with each division step, the process ultimately yields a zero remainder, making the last non-zero remainder the gcd of \(a(x)\) and \(b(x)\). This demonstrates why the gcd method described is valid, known as the Euclidean Algorithm.

Key Concepts

Division Algorithm

The Division Algorithm is a key principle applied to both numbers and polynomials. In the context of polynomials, it states that you can divide a polynomial \(a(x)\) by another polynomial \(b(x)\), provided that \(b(x)\) has a lower degree than \(a(x)\). This division will give you a quotient polynomial \(q_1(x)\) and a remainder polynomial \(r_1(x)\). The equation looks like this:

• \(a(x) = b(x) q_1(x) + r_1(x)\)
• where \(0 \leq \deg(r_1(x)) < \deg(b(x))\)

This process helps in breaking down complex polynomials into smaller, more manageable components, laying the foundational groundwork for finding the greatest common divisor.With each division step, the remainder's degree is always less than the divisor's, ensuring the process can eventually be concluded.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) of two polynomials is a crucial concept. It refers to the highest degree polynomial that divides two given polynomials without leaving any remainder. In essence, when we determine the GCD of two polynomials \(a(x)\) and \(b(x)\), we are identifying a polynomial that can evenly "fit" into both \(a(x)\) and \(b(x)\).

• If \(d(x)\) is the GCD, that means both \(a(x)\) and \(b(x)\) can be expressed as multiples of \(d(x)\).
• This ensures that \(d(x)\) is common to both, verifying that there are no larger polynomials that meet these conditions.

The GCD is essential in simplifying polynomial expressions and solving equations involving polynomials. Understanding the GCD helps in determining the "core" polynomial structure shared by both \(a(x)\) and \(b(x)\).

Common Divisors in Polynomials

A common divisor for polynomials \(a(x)\) and \(b(x)\) is a polynomial \(d(x)\) that divides both without leaving a remainder. Identifying common divisors is critical for simplifying polynomials and solving polynomial problems effectively.When you divide \(a(x)\) by \(b(x)\) and obtain \(r_1(x)\), any common divisor \(d(x)\) for \(a(x)\) and \(b(x)\) will also divide \(r_1(x)\). This is derived from the property that divisors of polynomials must divide any linear combination of them.

• For any two polynomials \(f(x)\) and \(g(x)\) if \(d(x)\) divides both \(f(x)\) and \(g(x)\), then \(d(x)\) divides \(Af(x) + Bg(x)\) for any polynomials \(A(x)\) and \(B(x)\).

By this property, \(d(x)\) divides the remainder \(r_1(x) = a(x) - b(x) q_1(x)\). Understanding this helps reinforce the concept that common divisors persist through the division process.

Polynomial Division Process

The Polynomial Division Process is similar in nature to the division process with integers, but it involves polynomials instead. We continuously divide the polynomial by the remainder from the previous division until a remainder of zero is achieved.

• Begin by dividing \(a(x)\) by \(b(x)\) to get a remainder \(r_1(x)\).
• Then, divide \(b(x)\) by \(r_1(x)\) to get \(r_2(x)\), and continue this process.

Each step of this division process involves a polynomial of a lower degree, ensuring that eventually, we will achieve a remainder of zero, indicating that we have found the GCD by the last non-zero remainder. The beauty of this repeatable process, known as the Euclidean Algorithm for Polynomials, lies in its efficiency and reliability in finding the GCD. This makes it an invaluable tool in polynomial mathematics and helps students understand the intricate relationships between polynomial components.