Let \(\theta\) be an algebraic number with minimal polynomial \(F\) over \({\mathbb Q}\). Let \(K={\mathbb Q}(\theta)\) and \({\mathcal O}_{K}\) be its ring of integers. The main result of the paper describes the roots \(\theta\) of trinomials \(F(x)=x^n+ax+b \in {\mathbb Z}[x]\) for which \({\mathcal O}_{K}={\mathbb Z}[\theta]\). The answer is given in terms of \(a,b\) and the prime divisors of the discriminant \(D_F\) of \(F\). In particular, for the root \(\theta\) of \(F(x)=x^n-x-1\) we have \({\mathcal O}_{K}={\mathbb Z}[\theta]\) iff \(|D_F|=n^n-(n-1)^{n-1}\) is square-free.