Problem 11 - Algebra: Pure and Applied

Question

Find a greatest common divisor \(d\) of \(a\) and \(b\) in the indicated Euclidean domain, and express \(d=u a+v b\). $$ a=4+7 i \quad b=8-i \quad \text { in } \mathbb{Z}[i] $$

Step-by-Step Solution

Answer

The greatest common divisor is \(1\) and can be expressed as \(1 = -i(-5)\).

Step 1: Calculate Norms

First, compute the norms of the given complex numbers \(a = 4 + 7i\) and \(b = 8 - i\) in the Gaussian integers \(\mathbb{Z}[i]\). The norm of a complex number \(z = x + yi\) is given by \(N(z) = x^2 + y^2\). Thus, \(N(a) = 4^2 + 7^2 = 16 + 49 = 65\) and \(N(b) = 8^2 + (-1)^2 = 64 + 1 = 65\).

Step 2: Perform Division in \(\mathbb{Z}[i]\)

Perform the division of \(a\) and \(b\) using division in the Gaussian integers. Divide \(a = 4 + 7i\) by \(b = 8 - i\). The division can be calculated as \(\frac{a}{b} = \frac{(4 + 7i)(8 + i)}{65}\) (since the denominator's norm is 65). This results in \(\frac{37 + 60i}{65} \approx 0.569 + 0.923i\). Convert this to a Gaussian integer by taking the nearest integer pair, which gives quotient \(q = 1 + i\).

Step 3: Compute Remainder

Calculate the remainder \(r\) when \(a\) is divided by \(b\): \(r = a - qb = (4 + 7i) - (1 + i)(8 - i)\). Perform the distribution: \((1 + i)(8 - i) = 8 + (-i) + 8i - i^2 = 8 + 7i + 1 = 9 + 7i\). Thus, \(r = (4 + 7i) - (9 + 7i) = -5\). Since the remainder is a Gaussian integer, it doesn't have an imaginary part.

Step 4: Iterate to Find GCD

Now apply the Euclidean algorithm between \(b\) and \(r\). But here the result is already evident since the remainder \(r = -5\) is real. Check that divisors of \(b\) include \(b\) and norm, the norm of \(b\) and check for division possibilities. Calculate greatest common divisor \(\text{gcd}(8 - i, -5) = 1\), the simplest integer.

Step 5: Express GCD as Linear Combination

Using the Euclidean algorithm outputs, compute the gcd's expression. From \(a - qb = -5\) and considering iteration till gcd finality in essence given conditions align for iterative pursuit until the remainder reaches one. Consequently, express \(-5 = (4 + 7i) - (1 + i)(8 - i)\), solving back using gcd as \(1 = (-i)(-5)\). Choose coefficients \(u = -i\) and \(v = 0\).

Key Concepts

Understanding the Euclidean Algorithm

The Euclidean algorithm is a classic method used for finding the greatest common divisor (GCD) of two integers, and it can be extended to Gaussian integers as well. In the world of complex numbers, specifically Gaussian integers \(\mathbb{Z}[i]\), the Euclidean algorithm helps us find the GCD by using a series of divisions.
The process involves repeated division steps where we obtain a quotient and a remainder, just like with real numbers. The algorithm continues until the remainder is zero. At this point, the divisor from the last non-zero remainder step is our GCD.
To make calculations easier:

• Perform divisions precisely, ensuring both the quotient and remainder are Gaussian integers.
• Iterate using the result of each division until obtaining a zero remainder.
• The final non-zero remainder is the GCD.

In the exercise, we divided complex number \(a = 4 + 7i\) by \(b = 8 - i\), ensuring that both the quotient and remainder were valid within this integer domain, leading us to eventually identify the GCD.

What is the Greatest Common Divisor?

The greatest common divisor (GCD) is an essential concept in number theory. For any two numbers, their GCD is the largest integer that divides both numbers without leaving a remainder. When working with Gaussian integers, this idea extends to finding the greatest Gaussian integer that divides both numbers.
To grasp this:

• The GCD in the domain of Gaussian integers may not be a simple integer but rather a Gaussian integer.
• In the exercise, since the norm comparisons confirmed no smaller common divisors, the gcd of the Gaussian integers \(4 + 7i\) and \(8 - i\) is found iteratively in their Euclidean algorithm resolution.
• Once identified, it can also be represented as a linear combination.

Every GCD problem steps move towards expressing the combination of the original numbers in such a way that highlights this common divisor, even as the complexity of the numbers increases with imaginary units.

Expressing as a Linear Combination

To express a number as a linear combination is to write it as a sum of multiples of given numbers. In this context, it means expressing the GCD as a combination of the original numbers, where Gaussian integers are coefficients.
For instance, if you find your GCD of two numbers \(a\) and \(b\) is \(d\), it should always be possible to express \(d = ua + vb\), where \(u\) and \(v\) are Gaussian integers.
The process involves back-substituting from the results obtained through the Euclidean algorithm. Here’s the approach used:

• Continue using the previous division remainders and quotients till the simplified form comes into light.
• Use the steps of division within the Euclidean algorithm to derive suitable coefficients \(u\) and \(v\).
• Apply these calculations to solve for the linear combination as back-solving takes place up the division chain.

In our problem, the GCD was expressed using coefficients derived through the Euclidean steps, establishing it in a linear equation format.

Complex Number Norms in Gaussian Integers

The norm in Gaussian integers is essential for applying the Euclidean algorithm. The norm of a complex number \(z = x + yi\) in Gaussian integers \(\mathbb{Z}[i]\) is calculated as \(N(z) = x^2 + y^2\).
The norm determines how we handle division and gives us a measure of the 'size' of a complex number within the Gaussian plane. It helps guide the Euclidean strategy and confirms divisor iteration.
Key observations:

• The norm is always a non-negative integer, helping classify which residues are possible as divisors.
• During the exercise, we calculated norms of both given numbers to check for fairness in division operations.
• Having norms of equal value highlighted the potential straightforwardness of deducing the GCD.

Understanding and calculating norms are integral in not only finding the GCD but also very crucial in performing any meaningful operations within the realm of Gaussian integers.