Find all elements of our ring R that have norm 1 Show that n

Find all elements of our ring R that have norm 1. Show that no element has norm 2 or 3.

Solution

a) This element is clearly non-zero, and not a unit. If it is a product of two

non-units, then its norm is a product of two integers of absolute value greater than

1, a contradiction

   b) for this we will assume that P is the principal

We need to show that

P s not principal, since

P^2= (2) (easy to check by direct

computation) its order in the class group divides 2, Now P has one generator , then the norm of this generator is 2

so we can say in the equation it has no principal

it means if it has no principal with elements it has no norm

so we orove for 3   similarly