Integral Domains and Fields

If \(i^2 = -1\), then the set \({\mathbb Z}[ i ] = \{ m + ni : m, n \in {\mathbb Z} \}\) forms a ring known as the Gaussian integers. \(\mathbb Z[i]\) the Gaussian integers It is easily seen that the Gaussian integers are a subring of the complex numbers since they are closed under addition and multiplication. Let \(\alpha = a + bi\) be a unit in \({\mathbb Z}[ i ]\). Then \(\overline{\alpha} = a - bi\) is also a unit since if \(\alpha \beta = 1\), then \(\overline{\alpha} \overline{\beta} = 1\). If \(\beta = c + di\), then \[\begin{aligned}\end{aligned}\]. Therefore, \(a^2 + b^2\) must either be \(1\) or \(-1\); or, equivalently, \(a + bi = \pm 1\) or \(a + bi = \pm i\). Therefore, units of this ring are \(\pm 1\) and \(\pm i\); hence, the Gaussian integers are not a field. We will leave it as an exercise to prove that the Gaussian integers are an integral domain.

The set of matrices \[\begin{aligned}\end{aligned}\] with entries in \({\mathbb Z}_2\) forms a field.

The set \({\mathbb Q}( \sqrt{2}\, ) = \{ a + b \sqrt{2} : a, b \in {\mathbb Q} \}\) is a field. The inverse of an element \(a + b \sqrt{2}\) in \({\mathbb Q}( \sqrt{2}\, )\) is \[\begin{aligned}\end{aligned}\].