Surjective Ring Homomorphism

Hi there!

This is my first time posting here so apologies if this is in the wrong section or there's already a post about this but I have looked everywhere to try and get a solution to this.

The question I'm stuck on is:

Let θ : Q[X] → Q(√3) be the map defined by θ(a0 + a1X + ... + anXn) = (a0 + a1√3 + ... + an(√3)n).

Show that θ is a surjective ring homomorphism.

Prove that Ker θ = (X^2 − 3)Q[X].

Deduce that the factor ring Q[X]/Ker θ is isomorphic to Q(√3).

The notes I have on this aren't particularly useful though I do believe that I need to use the first isomorphism theorem for the last part. The first part however I can't seem to find anything like in any of the books or websites I have looked at.

Thanks in advance for any help you can provide.