Let , where a and b are integers. Define the norm of α, denoted by Ν (α), as N (α) = a2 +...

Let , where a and b are integers. Define the norm of α, denoted by Ν (α), as N (α) = a2 + 5b2.

Use an argument similar to that in Exercise 1 to show that 3 is a prime number of the form .

Exercise 1

Let , where a and b are integers. Define the norm of α, denoted by Ν (α), as N (α) = a2 + 5b2.

A number of the form  is prime if it cannot be written as the product of numbers α and β, where neither αnor β equals ±1. Show that the number 2 is a prime number of the form . (Hint: Start with N(2) = Ν(αβ), and use: Exercise 2.)

Exercise 2

Let , where a and b are integers. Define the norm of α, denoted by Ν (α), as N (α)= a2 + 5b2.

Show that if  and , where a, b, c, and d are integers, then Ν (αβ) = Ν(α)Ν(β).