Prove that if p is a prime in Z that can be written in the...

Prove that if $p$ is a prime in $Z$ that can be written in the form $a^{2}+b^{2}$, then $a+b i$ is irreducible in $Z[i]$. Find three primes that have this property and the corresponding irreducibles.

Key Concepts

Representation of Primes as Sums of Two Squares

A classical result in number theory states that a prime number in the integers can be represented as a sum of two squares if and only if it is 2 or congruent to 1 modulo 4. When a prime p is expressed as a² + b², the corresponding Gaussian integer a + bi has norm p, and the property of being expressible as a sum of two squares directly plays a role in establishing its irreducibility in Z[i].

Gaussian Integers

Gaussian integers are complex numbers of the form x + yi where x and y are integers. They form a ring, denoted by Z[i], which shares many algebraic properties with the ordinary integers but also exhibits features unique to complex numbers. Concepts like divisibility, gcd, and factorization are extended to Z[i], making it a fundamental object in algebraic number theory.

Norm in Gaussian Integers

The norm function in the Gaussian integers is defined for an element x + yi as N(x + yi) = x² + y². This function is particularly significant because it maps Gaussian integers to non-negative integers and serves as a tool to measure the 'size' of an element. The norm is multiplicative, meaning that N(zw) = N(z)N(w), a property that is crucial in analyzing factors and irreducibility within Z[i].

Irreducibility in Integral Domains

In any integral domain, an element is called irreducible if it is not a unit and it cannot be expressed as a product of two non-units. For the Gaussian integers, showing the irreducibility of an element like a + bi often involves demonstrating that any factorization would involve factors whose norms are trivial (i.e., units), using the connection between the norm of an element and its potential factors.

Multiplicativity of the Norm

The multiplicativity of the norm in the Gaussian integers implies that the norm of a product equals the product of the norms of the factors. This property is instrumental in proving irreducibility: if an element has a prime norm (in Z), then any nontrivial factorization would lead to a contradiction since the norms of the factors must multiply to give the prime norm, forcing one of the factors to be a unit.

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