Problem: Let $K$ be a field and let $p(x)$ be a non-zero polynomial of $K[x]$.
Suppose that deg($p(x)$)$=1$, show that $p(x)$ is irreducible.
Thoughts: My definition of an irreducible element $p$ in an integral domain is that it is non-zero, not a unit, and that it's only divisors are the associates of $1_R$ and $p$.
Clearly $p(x)$ is neither zero or a unit since it has a coefficient of $x^1$. I am not sure how to proceed here, although my intuition is to apply the euclidean algorithm of division of polynomials. Any hints or insight much appreciated.