Question

Show that in the domain of complex integers of the form \(a+b \sqrt{-5}\), the integers \(2,3,-2+\sqrt{-5},-2-\sqrt{-5}, 1+\) \(\sqrt{-5}, 1-\sqrt{-5}\) are all irreducible.

Short answer

Answer: Yes, all the given complex integers are irreducible in the domain of complex integers of the form \(a+b\sqrt{-5}\).

Step-by-step solution

Define the norm function

To check irreducibility, we will use a norm function. In the domain of complex integers of the form \(a+b\sqrt{-5}\), the norm function is defined as: \(N(a+b\sqrt{-5}) = (a+b\sqrt{-5})(a-b\sqrt{-5}) = a^2 + 5b^2.\)

Check irreducibility of 2

Let us assume that \(2 = (a+b\sqrt{-5})(c+d\sqrt{-5})\). Then, applying the norm function to both sides, we have \(4=N(2)=N(a+b\sqrt{-5})N(c+d\sqrt{-5}) \Rightarrow N(a+b\sqrt{-5})\) and \(N(c+d\sqrt{-5})\) must be equal to either 1 or 4. If the norms are both equal to 1, then this would mean \((a+b\sqrt{-5})(c+d\sqrt{-5})\) is a unit, which is a contradiction. The only other possibility is that one of the factors has a norm of 4, but there is no complex integer of the specified form with a norm of 4. Thus, 2 is irreducible.

Check irreducibility of 3

Similarly, let's assume that 3 can be expressed as a product of two complex integers of the given form. Then we would have: \(3 = (a+b\sqrt{-5})(c+d\sqrt{-5})\). Applying the norm function, we obtain \(9=N(3)=N(a+b\sqrt{-5})N(c+d\sqrt{-5}) \Rightarrow N(a+b\sqrt{-5})\) and \(N(c+d\sqrt{-5})\) must be equal to either 1 or 9. But once again, we can't find any complex integer of the required form with a norm of 9. So, 3 is irreducible.

Check irreducibility of the remaining elements

The same procedure can now be applied to the remaining elements in the list, and we will find that they are irreducible as well. For example, let's consider \(-2+\sqrt{-5}\): Suppose for contradiction that \(-2+\sqrt{-5} = (a+b\sqrt{-5})(c+d\sqrt{-5})\). Then using the norm function: \(N(-2+\sqrt{-5}) = N(a+b\sqrt{-5})N(c+d\sqrt{-5})\) \(9 = N(a+b\sqrt{-5})N(c+d\sqrt{-5})\) In this case, we can't find any complex integers of the required form whose norms multiply to 9, thus proving that \(-2+\sqrt{-5}\) is irreducible. Similarly, we can prove the irreducibility of the remaining elements. In conclusion, all the given complex integers are irreducible in the domain of complex integers of the form \(a+b\sqrt{-5}\).

Key concepts

Norm Function

The norm function is a valuable tool in mathematics, especially when dealing with complex integers. In the context of complex integers of the form \(a + b\sqrt{-5}\), the norm function helps us understand the size or magnitude of these numbers.
The norm function, denoted here as \(N(a + b\sqrt{-5})\), is calculated as: \[N(a+b\sqrt{-5}) = (a+b\sqrt{-5})(a-b\sqrt{-5}) = a^2 + 5b^2.\] This formula is derived from multiplying \((a + b\sqrt{-5})\) by its conjugate \((a - b\sqrt{-5})\), which eliminates the imaginary part.

The result is always a non-negative integer, which is useful when checking for irreducibility. If a complex integer has a norm that cannot be expressed as the product of smaller norms of integers in the same domain, it is considered irreducible. This concept is crucial in the context of the given exercise.

Complex Numbers

Complex numbers are numbers that have both a real and an imaginary part, typically written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2 = -1\).

However, in the domain of the exercise, we consider complex integers of a specific form, \(a + b\sqrt{-5}\). These types of numbers extend the concept of complex numbers by incorporating a square root of a negative integer, which changes the structure of the numbers.

Understanding complex numbers involves:

• Knowing that the "+" in \(a + bi\) or \(a + b\sqrt{-5}\) does not imply simple arithmetic addition but a unique combination of real and imaginary parts.
• Recognizing that the imaginary part creates a plane of numbers called the complex plane, allowing us to visualize and manipulate these numbers geometrically.

The awareness of their geometric and conventional algebraic properties aids in deeper understanding and manipulation in advanced calculations.

Algebraic Integers

Algebraic integers are a more generalized form of integers that include solutions to some polynomial equations, specifically those with integer coefficients and leading coefficient 1. In simpler terms, these are numbers that, when plugged into particular polynomial equations, result in zero.

In the case of complex integers like \(a + b\sqrt{-5}\), we are dealing with algebraic integers since these numbers solve polynomial equations of the form \(x^2 + 5 = 0\) if \(x = \sqrt{-5}\). These complex integers comply with the same foundational properties as regular integers, such as divisibility and factorization, but in a broader and more abstract sense.

Algebraic integers play a critical role when examining irreducibility since they follow the rules of arithmetic within given domains, and determining whether numbers can be factored further tests their structure and properties.

Mathematical Proof

Mathematical proofs are logical arguments verifying the truth of a mathematical statement. They are crucial in confirming the properties of complex numbers, such as irreducibility.

A proof involves several steps:

• Statement: Clearly define the problem or property under investigation (e.g., proving the irreducibility of a complex integer).
• Assumptions: Declare any initial assumptions, often derived from known mathematical facts or prior proven theorems.
• Logical Argument: Construct a sequence of logical deductions that bridge the assumptions to the conclusion.
• Conclusion: Clearly summarize the result as demonstrated through the argumentation.

In the complex integers of the form \(a+b\sqrt{-5}\), mathematical proofs help in verifying irreducibility by showing that no proper divisors exist. By applying the norm function and considering all possibilities, we establish facts with certainty, thus unveiling intricate mathematical truths.