# Multiplicative group of a finite field is cyclic

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## Statement

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

## Related facts

### Consequences

- Note in particular that any finite field has order a prime power, say for a prime number and positive integer . The multiplicative group is therefore a cyclic group of order .
- Since the multiplicative group is cyclic, there are many choices for the generator of the group. Here, is the Euler totient function.
- Multiplicative group of a finite prime field is cyclic (see also classification of natural numbers for which the multiplicative group is cyclic)

- For , any generator of the multiplicative group is also a primitive element for the field of elements as an extension of its prime subfield (of 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 . The multiplicative group has generators, whereas the field has primitive elements.
- Every finite division ring is a field

## Facts used

## Proof

The statement follows directly from fact (1).