Mums respond well to being divided, but you should wait until spring to do so. For now, leave the mums as-is. Since you live in a climate with a true winter, I would not recommend cutting the plants back this fall. Simply let them die back. After the ground has started to freeze, you can add mulch (straw, leaves, pine branches - whatever you have) to protect them for the winter. The mulch stays in place better when the plant has not been cut back.
In the spring, remove the mulch. I cut mine back as soon as new growth starts to show. That is also when you can divide them. Dig up each plant with a sharp spade. You should be able to see clumps within the root mass. Pull or cut the clumps of roots apart, and replant each clump separately with the crown right at or just below the soil level. Water them well.
In my climate, it is a bit hit-or-miss as to whether hardy mums make it through the winter at all. About half of them seem to make it, regardless of where they are planted in the yard.