This question is from Dummit and Foote Abstract Algebra pg 293 #5.
Let where n is a square free integeter greater than 3.
(a) PRove that are irreducable in R.
(b) Prove that R is not a U.F.D Conclude that the quadratic inter ring is not a U.F.D for
Show that is not prime.
(c) Give an explicit ideal in R that is not principal. Hint: Use (b) and cosider a maximal ideal containing the non prime ideal
For part (a) I have shown that none of the of them are units i.e they don't have a norm of 1 and they are non zero. so the only thing I have left to do is show if they do factor one of the factors is a unit
so for 2
suppose that there is a factorization then there exits such that
Since the norm is a multiplicitive function I get
Since a and b are not units and n(a) and N(b) are integers greater than 1
but since n > 3 and there does not exista an integer such that
There for 2 is irreducable.
This is where I am stuck.
I don't see how to generalize this argument.