Multiplicative group of a finite prime field is cyclic

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Statement

In the language of modular arithmetic

Let p be a prime number. Then, the multiplicative group modulo p is a cyclic group of order p-1. In other words, it is isomorphic to the group of integers modulo p - 1.

A generator for this multiplicative group is termed a primitive root modulo p. 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

For commutative rings

Noncommutative analogues

Examples

The prime 2

The multiplicative group modulo 2 is the trivial group, so this is not an interesting case.

The prime 3

The multiplicative group modulo 3 is of order two, and the element 2 is a primitive root in this case.

The prime 5

The multiplicative group modulo 5 is of order four. The element 2 is a primitive root. The powers of 2 include all elements: 2^0 = 1, 2^1 = 2, 2^2 = 4, 2^3 = 3.

3 is also a primitive root. Its powers are 3^0 = 1, 3^1 = 3, 3^2 = 4, 3^3 = 2.

The prime 7

The multiplicative group modulo 7 is of order six. 2 is not a primitive root: it has order 3, and its powers only include 1,2,4. On the other hand, 3 is a primitive root, with 3^0 = 1, 3^1 = 3, 3^2 = 2, 3^3 = 6, 3^4 = 4, 3^5 = 5.

Facts used

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

Proof

The proof follows directly from fact (1).