What is the role of the nucleus in a eukaryotic cell?

The nucleus serves as the control center of a eukaryotic cell, primarily responsible for storing and managing the cell's genetic material, which is organized into chromosomes. It contains the cell's DNA, which carries the instructions required for the development, functioning, growth, and reproduction of the organism. Within the nucleus, processes such as transcription occur, where the DNA is transcribed into messenger RNA (mRNA) that later guides protein synthesis in the cytoplasm. This vital function makes the nucleus essential for maintaining the cell's genetic information and ensuring that it is accurately replicated during cell division and expressed as needed.

In contrast, other options relate to different cellular functions: energy production is primarily associated with organelles like mitochondria, photosynthesis occurs in chloroplasts in plant cells, and protein degradation is mainly carried out by proteasomes and lysosomes. Hence, the nucleus's unique role in genetic material storage is what distinctly qualifies it as the correct choice in this context.