I know little of cacti. This is a prickly long column (9" tall?). If I'd make a cross-section cut and looking down on the top it probably looks like shape of a star. But base looks dark and shrunken. Can I save the broken piece? How? It also looks a little scorched from the sun.
Cactus Broke Off Base
Certified GKH Gardening Expert
There are a number of cacti listed in this article: https://www.gardeningknowhow.com/ornamental/cacti-succulents/scgen/exotic-showstoppers-beautiful-succulents.htm
You may be able to recognize your plant from one of the photos.
Usually, the end of a broken piece of cactus can easily be rooted. If you have new or "still healthy" growth on the base of your plant, you could just leave it in one piece and concentrate on making your plant happy with the proper lighting, water and soil content. Or, if you're able to identify the plant, you could try propagating new plants with cuttings off of the base. I would probably try keeping it whole first, and start pruning off pieces for cuttings as a last resort.
This article may also help you: https://www.gardeningknowhow.com/ornamental/cacti-succulents/starfish-flower/growing-starfish-flowers.htm