Multiplicative group of a finite field is cyclic
From Groupprops
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).