Why is it that R = {a + bsqrt(2): a, b belong to Z} is a domain, but R = { (1/2)(a + bsqrt(2)): a, b belong to Z} isn't?
The two seem to be abelian groups for but take a closer look at the stability under multiplication.
For example, in the second belongs to it, but what about the product ?
If there are no such that then is not stable under multiplication and it cannot be a ring.
I understand proving cases where R is not a domain perfectly now, but I am still confused about how to prove cases where R is a domain. Another question I was assigned was to use the fact that c = (1/2)(1 + sqrt(-19)) is a root of x^2 - x + 5 to prove that R = {a + bc: a,b belong to Z} is a domain and I don't know where to start.
and as you defined.
A good way to prove something is a ring is to show it is a subring of some ring you already know; here one can think of , since .
Proving that is a subgroup of is quite easy.
because
The most tricky (and here the last) part is to show that is closed under multiplication. So take two elements of let's say and and consider their product: (we want to prove it belongs to )
As we're working in which is a commutative ring, we can use the properties of addition and multiplication.
by distributivity of over
by commutativity of
by commutativity of
by distributivity of over
because c root of means
by distributivity of over and commutativity of and
Note that implies that and belong to i.e. and we're done.
That proof of the closure of R under multiplication is quite detailled, that's why it can seem long...
I nearly forgot, you wanted to show it is a domain, i.e. the product of two non-nul elements is a non-nul element. Well a subring of a domain is a domain, so since is a domain, we immediately have that is also a domain.
Using a similar (and even easier) proof, you can show that is a domain too!