Abstract algebra

1. Jun 21, 2015

NanoMath

1. The problem statement, all variables and given/known data

Hello guys
So I have the following problem, given the mapping above I have to check weather it's ring homomorphism, and
maybe monomorphism or epimorphism.

3. The attempt at a solution

So the mapping is obviously well defined, and I have proven it's homomorphism, and it's obviously not monomorphism because a polynomial P(x)= 5 - x2 is in the kernel so kernel is not trivial.
I am not sure how to prove if the function is surjective or not, obviously if the codomain were integers for every integers C , I could just use constant function p(x) = C and function would be surjective.

2. Jun 21, 2015

geoffrey159

It is just an intuition, but I doubt there are polynomials of $\mathbb{Z}[X]$ such that $p(\sqrt{5}) \in \mathbb{Q}-\mathbb{Z}$.

3. Jun 22, 2015

Zondrina

Showing the map is a homomorphism shouldn't be too difficult. Simply show the map satisfies the properties of a homomorphism.

The rest of the question is asking if the map is also a bijection, or something more specific. Try applying the first isomorphism theorem if you know it, and use the fact that $x^2 - 5$ is the minimal polynomial.

4. Jun 22, 2015

Dick

Isn't it pretty obvious that the range is contained in the set $a+b \sqrt{5}$ where $a$ and $b$ are integers? Why isn't that all of $R$?

5. Jun 23, 2015

NanoMath

I managed to show that function is not surjective with the hint that every element in the range is of the form $a+b \sqrt{5}$ because for example $\sqrt{2}$ doesn't get hit by any element in domain. Is it also valid argument that function can't be surjective because $\mathbb{R}$ is uncountable whilst $\mathbb{Z}[X]$ is countable?

6. Jun 23, 2015

Dick

Both of those arguments are good. The second makes it obvious if you know cardinality.