We know that f(x) is irreducible in K[x] if and only if f(x+a) is irreducible in K[x] ... Wherer K is a field and f(x) is a polynomial over K
My question is why does K has to be a field, i know all the non zero elements in K are invertible, but why can't this rule be true in any ring?? Please help
You don't actually need $K$ to be a field. This works because the map $K[x] \longrightarrow K[x]$ via $x \mapsto x+a$ is an isomorphism of rings, and $K$ need not be a field for this to be an isomorphism.