In the ring , determine can the following elements be decomposed into primary factors and is the decomposition unique:
.
I can't find anything in my notes on this, could someone please help me out, or maybe direct me to some online notes where it is explained?
Thank you!
Define . Notice that if and only if is a unit in this ring.
Also notice that .
If , a prime, then must be irreducible in this ring. Because if it was then we can write for non-units and so by taking norms, but the factors non-trivially since for they are non-units. This is a contradiction.
Since it means it is irreducible.
Since it means it is a unit.
I see that NonCommAlg already replied with the case involving .