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Every finite division ring is commutative, hence a field.
Wedderburn's little theorem: Every finite division ring is commutative. Equivalently, every finite division ring is a field.
A standard approach studies the finite group of units, i.e. the group of units, and compares its conjugacy structure with the center of the ring using the class equation.
A division ring (or skew field) is a unital ring with such that every nonzero element of is a unit (equivalently, is a group under multiplication, i.e. the group of units equals all nonzero elements).
A field is a commutative ring with such that every nonzero element is a unit (equivalently, every has a multiplicative inverse).
Class Equation. Let be a finite group. For , let the conjugacy class be
and let the centralizer be
Then . If denotes the center of and are representatives of the distinct conjugacy classes contained in , then