Define the norm N( ) for an element . Show that any common divisor of and 2 in the ring is a unit.
I know that
N( and that
N(2)= 4
But how do you show that any common divisor is in the ring?
N(xy)=N(x)N(y) where .
There are three things to consider:
1. N(x) >= 0
2. if N(x) = 1, then x is a unit.
3. As Swlabr said, there is no element x such that N(x)=2.
If the common divisor of your example, let's say k, has the norm N(k)=1, then k is the unit in your ring. Can you conclude now?
No. Any common divisor will have norm 1, and is a unit. This may be in your notes (the is a unit bit). Otherwise, it is good exercise to prove it.
You need to prove that if a|2 and a| then N(a)=1. To do this, you must notice that as and so if a|2 and a| then N(a)|4. There are only three numbers which divide 4. They are 1, 2 and 4.
If N(a)=1 then is a unit.
N(a)=2 cannot happen, you must prove this.
If N(a)=4 then you need to prove that cannot divide both ring elements. To do this, notice that there must exist such that and . Is this possible?