# Multiplicative group of a finite prime field is cyclic

## Contents

## Statement

### In the language of modular arithmetic

Let be a prime number. Then, the multiplicative group modulo is a cyclic group of order . In other words, it is isomorphic to the group of integers modulo .

A generator for this multiplicative group is termed a primitive root modulo . While the theorem states that primitive roots exist, there is no procedure or formula known for obtaining a primitive root.

### In the language of fields

The multiplicative group of a prime field is a cyclic group.

## Related facts

### For fields

- Multiplicative group of a field implies at most n elements of order dividing n
- At most n elements of order dividing n implies every finite subgroup is cyclic
- Multiplicative group of a field implies every finite subgroup is cyclic
- Multiplicative group of a finite field is cyclic
- Classification of fields whose multiplicative group is uniquely divisible
- Classification of fields whose multiplicative group is locally cyclic

### For commutative rings

### Noncommutative analogues

## Examples

### The prime 2

The multiplicative group modulo is the trivial group, so this is not an interesting case.

### The prime 3

The multiplicative group modulo is of order two, and the element is a primitive root in this case.

### The prime 5

The multiplicative group modulo is of order four. The element is a primitive root. The powers of include all elements: .

is also a primitive root. Its powers are .

### The prime 7

The multiplicative group modulo is of order six. is *not* a primitive root: it has order , and its powers only include . On the other hand, is a primitive root, with .

## Facts used

## Proof

The proof follows directly from fact (1).