- Prove that if \(R\) is an integral domain, then \(R[t]\) is again an integral domain.
Solution
Let \[ f(t)=a_nt^n+\cdots+a_0,\qquad g(t)=b_mt^m+\cdots+b_0 \] be nonzero polynomials in \(R[t]\), with \(a_n,b_m\neq0\). The coefficient of \(t^{n+m}\) in \(fg\) is \(a_nb_m\). Since \(R\) is an integral domain, \(a_nb_m\neq0\). Hence \(fg\neq0\).
Therefore \(R[t]\) has no nonzero zero divisors, so it is an integral domain. In particular, \[ \deg(fg)=\deg f+\deg g \] for nonzero \(f,g\in R[t]\).