The nucleus controls protein synthesis in a cell's cytoplasm in the sense that the genetic codes contained in the nucleus determine the sequence of amino acids that are assembled to build the proteins. This is the only aspect of protein assembly directly controlled by the nucleus, while other mechanisms and structures in the cell are responsible for detecting which proteins are needed. The nucleus serves as a protein code database.
The cell nucleus is an assembly of coiled chromosomes surrounded by a membrane. Chromosomes, in turn, are groups of large numbers of genes, each of which is composed of many nucleotides. Each sequence of three nucleotides is a codon, which indicates a single amino acid. However, when they are coiled up, chromosomes are inactive and unreadable by cellular machinery. Chemical sensors in the cell must be activated, indicating a need for one protein or another, for the nucleus to begin working.
Once the chemical senses of the cell indicate a need for a protein, certain molecules travel to the nucleus. These molecules cause the necessary chromosome to uncoil, making it accessible for reading. The double helix of the DNA is split, exposing the needed nucleotides on one side, and then read, producing a reversed copy in the form of RNA. It is this RNA that is carried to the protein assembly machinery in the cytoplasm.