The question is for a module on Galois theory and I haven't a clue where to start:
Take $f \in \mathbb{K}[x]$ with $deg(f)>0$. Show that the map $\lambda \mapsto \lambda+<f>$ is an injective homomorphism $\mathbb{K} \rightarrow \mathbb{K}[x]/<f>$ and so $\mathbb{K}$ is isomorphic to its image in $\mathbb{K}[x]/<f>$.
This is where $<f>$ is the ideal generated by the polynomial $f$.
Your map is the composition of the inclusion homomorphism from $K$ to $K[x]$ with the projection homomorphism from $K[x]$ to $K[x]/\langle\lambda\rangle$. The composition of ring homomorphisms is a ring homomorphism.
A field has no nontrivial ideals, so a homomorphism from $K$ to a ring is either injective, or is zero. But your homomorphism takes $1$ to $1+\langle\lambda\rangle$ which is the zero element if and only if $\lambda$ is a factor of $1$ in $K[x]$.