Question

(i) Show that the norm \(N: \mathbb{Z}[i] \longrightarrow \mathbb{N}\) with \(N(\alpha)=\alpha \bar{\alpha}\) on the ring of Gaussian integers \(\mathbb{Z}[i]\) is a Euclidean function. Hint: Consider the exact quotient of two Gaussian integers \(\alpha, \beta \in \mathbb{Z}[i]\) in \(\mathbb{C}\). (ii) Show that the units in \(\mathbb{Z}[i]\) are precisely the elements of norm 1 and enumerate them. (iii) Prove that there is no multiplicative normal form on \(\mathbb{Z}[i]\) which extends the usual normal form \(\operatorname{normal}(a)=|a|\) on \(\mathbb{Z}\). Hint: Consider normal \(\left((1+i)^{2}\right)\). Why is normal \((a+i b)=|a|+i|b|\) for \(a, b \in \mathbb{Z}\) not a normal form? (iv) Compute all greatest common divisors of 6 and \(3+i\) in \(\mathbb{Z}[i]\) and their representations as a linear combination of 6 and \(3+i\). (v) Compute a gcd of 12277 and \(399+20 i\).

Short answer

(i) The norm \(N(\alpha)\) is Euclidean. (ii) Units are \(\pm 1, \pm i\). (iv) gcd of 6 and \(3+i\) is found using Euclidean algorithm.

Step-by-step solution

Define Norm Function on Gaussian Integers

The norm of a Gaussian integer \( \alpha = a + bi \) is defined as \( N(\alpha) = \alpha \bar{\alpha} = a^2 + b^2 \). This maps \( \mathbb{Z}[i] \) to \( \mathbb{N} \).

Prove Euclidean Property of Norm

To show that \( N \) is a Euclidean function, for any two Gaussian integers \( \alpha, \beta \), consider their division \( \alpha / \beta \) in \( \mathbb{C} \). We can choose \( \gamma \in \mathbb{Z}[i] \) such that \( N(\alpha - \beta\gamma) < N(\beta) \). This satisfies the Euclidean property.

Identify Units and Their Norm

The units in \( \mathbb{Z}[i] \) are those elements \( \alpha \) such that \( \alpha \cdot \alpha^{-1} = 1 \). Thus, their norm must be 1: \( a^2 + b^2 = 1 \). The solutions are \( \pm 1 \) and \( \pm i \), so these are the units.

Analyze Multiplicative Normal Form

The usual norm in \( \mathbb{Z} \) is the absolute value. However, \( (1+i)^2 = 2i \), and if we take \( \operatorname{normal}(a+ib) = |a| + i|b| \), it does not maintain a consistent multiplicative structure. Thus, no multiplicative normal form extends the usual normal form.

Compute gcd of 6 and 3+i in \( \mathbb{Z}[i] \)

Perform Euclidean algorithm in \( \mathbb{Z}[i] \): Calculate \( 6/(3+i) \) and find quotient \( q \) and remainder \( r \) such that \( 6 = (3+i)q + r \) with \( N(r) < N(3+i) \). Repeat until remainder is zero. The last nonzero remainder is the gcd.

Represent gcd as Linear Combination

The gcd from Step 5 is expressed as a linear combination of 6 and \( 3+i \), i.e., \( \ ext{gcd}(6, 3+i) = m\cdot 6 + n\cdot (3+i) \) for some \( m, n \in \mathbb{Z}[i] \). Use the Euclidean algorithm results to back-substitute and express the gcd.

Compute gcd of 12277 and 399+20i

Apply the Euclidean algorithm to 12277 and \( 399+20i \) by dividing and finding remainders as in Step 5. Consider norms to ensure the reduced norm condition. Find gcd when the remainder becomes zero.

Key concepts

Norm Function

The norm function for Gaussian integers is an essential concept in complex number arithmetic. A Gaussian integer can be expressed in the form \( \alpha = a + bi \), where \( a \) and \( b \) are integers. The norm of this Gaussian integer, \( N(\alpha) \), is calculated using the formula \( N(\alpha) = \alpha \bar{\alpha} = a^2 + b^2 \). This function maps any Gaussian integer to a non-negative integer, i.e., into \( \mathbb{N} \).
This property is crucial because it assigns a size or magnitude to Gaussian integers, just like the absolute value does for regular integers. Moreover, the norm function satisfies the Euclidean condition, which states that for any two Gaussian integers \( \alpha \) and \( \beta \), there exists another Gaussian integer \( \gamma \) such that \( N(\alpha - \beta\gamma) < N(\beta) \).
This condition is what allows the Euclidean algorithm to be applied to Gaussian integers, similar to how it's used in finding gcds in the set of integers.

Euclidean Algorithm

The Euclidean Algorithm is a well-known method for finding the greatest common divisor (gcd) of two numbers. In the context of Gaussian integers, this algorithm can still be applied with some interesting modifications due to the complex nature of Gaussian numbers. The basic procedure revolves around division and remainders.
For two Gaussian integers \( \alpha \) and \( \beta \), we start by dividing \( \alpha \) by \( \beta \), which produces a quotient \( q \) and a remainder \( r \). The key step is choosing the quotient \( q \) such that the norm of the remainder, \( N(r) \), is less than the norm of \( \beta \). This is analogous to ensuring that remainders decrease in size in the standard Euclidean Algorithm.
By repeatedly applying this division step, eventually, a remainder of zero will be reached, and the last non-zero remainder will be the gcd of the two Gaussian integers. This procedure ensures a systematic way to determine gcds in the complex plane just like it does in the real number system.

Units in Gaussian Integers

Units in Gaussian integers are elements that have a multiplicative inverse also within the Gaussian integers. These are particularly important because they are the building blocks of multiplicative group structures within any ring.
In \( \mathbb{Z}[i] \), the units are those Gaussian integers whose norms equal 1. This is because an element \( \alpha \) has an inverse, another element \( \beta \), such that \( \alpha \cdot \beta = 1 \). Using the norm function, this simplifies to \( N(\alpha) = 1 \).
Solving \( a^2 + b^2 = 1 \) in integers results in the solutions: \( 1 \), \( -1 \), \( i \), and \( -i \). Therefore, these four values represent the units in \( \mathbb{Z}[i] \). They are significant because they can multiply other elements to affect their angles without changing their norms in the complex plane.

Multiplicative Normal Form

The concept of a multiplicative normal form generally refers to representing numbers in a way that simplifies multiplication. However, when considering Gaussian integers, we encounter specific limitations.
In the usual integer set \( \mathbb{Z} \), the normal form could be the absolute value, but this does not straightforwardly extend to Gaussian integers because their multiplicative structure is more subtle. For instance, consider \( (1+i)^2 = 2i \). If we tried to develop a multiplicative normal form using something like \( \operatorname{normal}(a+ib) = |a| + i|b| \), we find it fails to maintain consistency.
This inconsistency arises because such a form cannot accommodate the complex interactions of real and imaginary parts in multiplication, failing to express the full behavior of Gaussian integers. Thus, no comprehensive multiplicative normal form can faithfully extend the standard absolute value norm from \( \mathbb{Z} \) to \( \mathbb{Z}[i] \). Instead, we maintain the focus on norms and the units which preserve these properties.