Partially Ordered Fields and Integral Domains

The article studies the division closed partial orders on fields that are algebraic over the field of rational numbers. In particular, the maximal partial orders are described using embeddings from the given field to the field of complex numbers. The \(O^{*}\) $O^{\ast }$ -fields that are not finite dimensional over \(\mathbb {Q}\) $\mathbb{Q}$ are studied in Section 2 and the \(n^{th}\) $n^{\mathit{th}}$ -root function over totally ordered fields is considered in Section 3.