Let \(K\) be a field. A polynomial \(p\in K[x]\) is irreducible if \(\deg p \geq 1\) and whenever \(p= qr\) with \(q,r\in K[x]\), \(q\) or \(r\) is constant.
Every polynomial of degree \(1\) is irreducible: if \(p=qr\), then \(\deg q+\deg r=1\), so one of \(q,r\) has degree \(0\).
When is a polynomial over a field irreducible?