Factor Each Of The Elements Below As A Product Of Irreducibles In Z[i], [Hint: Any Factor Of Aa Must appears to be the beginning of a problem involving factorization in the ring of Gaussian integers, Z[i]. Understanding how to factor elements within this structure requires an exploration of algebraic properties, units, norms, and irreducibility within Z[i]. This article aims to provide a comprehensive guide to factoring elements in Z[i] into irreducible elements, often called Gaussian primes, and to clarify the underlying principles involved in such factorizations.
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Introduction to Z[i]: The Ring of Gaussian Integers
The ring of Gaussian integers, denoted as Z[i], consists of all complex numbers of the form a + bi where a and b are integers, and i is the imaginary unit satisfying i² = -1. This ring extends the ordinary integers ℤ into the complex plane and possesses unique factorization properties similar to ℤ, but with notable differences due to the complex structure.
Properties of Z[i]
- Additive and Multiplicative Closure: For any a + bi, c + di in Z[i], their sum and product are also in Z[i].
- Units in Z[i]: Elements with multiplicative inverses also in Z[i] are called units. In Z[i], the units are precisely those elements with norm 1, i.e., a² + b² = 1. The units are:
- 1, -1, i, -i
- Norm Function: The norm N(a + bi) = a² + b², which maps elements of Z[i] to non-negative integers. The norm is multiplicative: N(zw) = N(z)N(w).
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Irreducible Elements in Z[i]
In Z[i], an element is irreducible if it cannot be factored into non-unit elements with smaller norm. These are often called Gaussian primes because they are the building blocks for all elements in Z[i].
Characterization of Gaussian Primes
- If a rational prime p remains prime in Z[i], it is called inert. For example:
- p ≡ 3 mod 4, p prime in ℤ, remains prime in Z[i].
- If p ≡ 1 mod 4, p factors into two Gaussian primes:
- p = π · π̄, where π and π̄ are conjugate Gaussian primes.
- The prime 2 factors as 2 = (1 + i)(1 - i), which are associates of each other, and both are Gaussian primes with norm 2.
- Rational primes p ≡ 3 mod 4 are Gaussian primes in Z[i].
- Rational primes p ≡ 1 mod 4 factor into Gaussian primes as p = (a + bi)(a - bi), with p = a² + b².
- 2 is special: 2 = (1 + i)(1 - i).
Factorization in Z[i]
The goal is to express any element in Z[i] as a product of units and irreducibles (Gaussian primes). The process involves:
- Calculating the norm.
- Factoring the norm into integers.
- Using the properties of rational primes to identify how they factor in Z[i].
- Expressing elements as products of Gaussian primes.
Key Steps in the Factorization Process
- Determine the norm of the element.
- Factor the norm into prime factors in ℤ.
- For each prime factor, determine if it remains prime in Z[i] or splits.
- Express the element as a product involving Gaussian primes.
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Step-by-Step Guide to Factor Elements in Z[i]
Let's explore how to factor specific elements in Z[i], illustrating the process with examples.
1. Factor a Simple Element: 5
- Norm: N(5) = 25.
- Factorization in ℤ: 25 = 5².
- Since 5 ≡ 1 mod 4, it factors in Z[i]:
- 5 = (2 + i)(2 - i), both Gaussian primes with norm 5.
- Conclusion: 5 = (2 + i)(2 - i), a product of two Gaussian primes.
2. Factor 7
- Norm: N(7) = 49.
- 7 ≡ 3 mod 4, remains prime in Z[i].
- Conclusion: 7 is a Gaussian prime itself.
3. Factor 2
- Norm: N(2) = 4.
- 2 factors as 2 = (1 + i)(1 - i).
- Both factors are Gaussian primes with norm 2.
4. Factor an element like 3 + 4i
- Norm: N(3 + 4i) = 3² + 4² = 9 + 16 = 25.
- 25 factors as 5², and since 5 factors into (2 + i)(2 - i), we look to express 3 + 4i as a product involving these factors.
- Express 3 + 4i as a product of Gaussian primes:
- Note that (3 + 4i) is associated with 5, since its norm is 25.
- Find factors:
- (3 + 4i) = (a + bi)(c + di), where the norms multiply to 25.
- Possible factorization:
- 3 + 4i = (1 + i)(2 + 3i), since:
- N(1 + i) = 2
- N(2 + 3i) = 4 + 9 = 13 (not 25)
- Alternatively, check directly:
- N(2 + i) = 5, which suggests that 3 + 4i might be associate to (2 + i)(1 + 2i):
- (2 + i)(1 + 2i) = 2(1 + 2i) + i(1 + 2i) = 2 + 4i + i + 2i² = 2 + 4i + i - 2 = (2 - 2) + (4i + i) = 0 + 5i.
- Not matching 3 + 4i.
- Instead, directly check if 3 + 4i is divisible by (1 + i):
- Divide: (3 + 4i)/(1 + i) = (3 + 4i)(1 - i)/(1 + i)(1 - i) = (3 + 4i)(1 - i)/2.
- Compute numerator:
- (3 + 4i)(1 - i) = 3(1 - i) + 4i(1 - i) = 3 - 3i + 4i - 4i² = 3 + i + 4(1) = 3 + i + 4 = 7 + i.
- Divide by 2: (7 + i)/2, which is not in Z[i], so not divisible by (1 + i).
- Therefore, 3 + 4i is a Gaussian prime itself (since its norm is 25, a prime in ℤ, and it does not factor further).
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Using Norms and Prime Factorization to Factor Elements
The key to factoring in Z[i] lies in the properties of the norm and how rational primes behave within the Gaussian integers.
Norms and Their Role
- The norm maps elements to non-negative integers.
- Factoring the norm into primes guides the factorization in Z[i].
- Since the norm is multiplicative, N(zw) = N(z)N(w), the factorization of N(z) into primes reflects the potential factorization of z.
Prime Factorization Strategy
- Compute N(z).
- Factor N(z) into prime factors in ℤ.
- For each prime p dividing N(z):
- If p ≡ 3 mod 4, then p remains prime in Z[i].
- If p ≡ 1 mod 4, then p factors into Gaussian primes.
- For p = 2, note its special factorization.
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Examples of Complete Factorizations
Let's look at some comprehensive examples illustrating the process:
Example 1: Factor 13
- Norm: N(13) = 169.
- 13 ≡ 1 mod 4, so it factors in Z[i]:
- 13 = (3 + 2i)(3 - 2i).
- Both 3 + 2i and 3 - 2i are Gaussian primes with norm 13.
