Abstract Algebra - define and find kernal of homomorphism from $\mathbb Q[x]$ to $\mathbb Q[\sqrt2]$

Question

The question asks you to find a monic irreducible polynomial $f$ so that $\mathbb Q[\sqrt2]$ is isomorphic to $\mathbb Q[x]/(f)$, so I was going to use the fundamental homomorphism theorem, and defined the homomorphism as $f(x)$ to $f(\sqrt2)$ (although I am finding it hard to prove that this is a homomorphism), and I am struggling to define the kernal.

Answer 1

Perhaps it is easier to define a map $φ : \mathbb{Q}[x]/(x^2 − 2)\rightarrow \mathbb{Q}(\sqrt{2})$ by $$ φ([ax + b]) = b + a\sqrt{2} $$ and to show that this is a surjective homomorphism with trivial kernel.