Abstract Algebra Homework Solution - Check Ring Homomorphism

[NanoMath]

Homework Statement

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.

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.

 

[geoffrey159]

I am not sure how to prove if the function is surjective or not

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

 

[STEMucator]

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.

 

[Dick]

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.

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 $R$?

 

[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?

 

[Dick]

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?

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