Multiplicative group of a finite field is cyclic

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Statement

Suppose F is a finite field. Let F^* denote the multiplicative group of F. Then, F^* is a cyclic group.

Related facts

Consequences

Other related facts

  • For r > 1, any generator of the multiplicative group is also a primitive element for the field of q elements as an extension of its prime subfield (of p elements). (A primitive element for a field extension is an element that, when adjoined to the smaller field, generates the larger field). However, not every primitive element is a generator of the multiplicative group. In fact, the number of generators of the multiplicative group could be substantially smaller than the number of primitive elements. For instance, consider the case p = 5, r = 2. The multiplicative group has \varphi(24) = 8 generators, whereas the field has p^2 - p = 25 - 5 = 20 primitive elements.
  • Every finite division ring is a field

Facts used

  1. Multiplicative group of a field implies every finite subgroup is cyclic

Proof

The statement follows directly from fact (1).