Primitive element (finite field)

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In field theory, a primitive element of a finite field GF(q) is a generator of the multiplicative group of the field. In other words, \alpha \in \mathrm{GF}(q) is called a primitive element if it is a primitive (q-1) root of unity in GF(q); this means that all the non-zero elements of \mathrm{GF}(q) can be written as \alpha^i for some (positive) integer i.

For example, 2 is a primitive element of the field GF(3) and GF(5), but not of GF(7) since it generates the cyclic subgroup of order 3 {2,4,1}; however, 3 is a primitive element of GF(7). The minimal polynomial of a primitive element is a primitive polynomial.

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Number of Primitive Elements[edit]

The number of primitive elements in a finite field GF(n) is φ(n - 1), where φ(m) is Euler's totient function, which counts the number of elements less than or equal to m which are relatively prime to m. This can be proved by using the theorem that the multiplicative group of a finite field GF(n) is cyclic of order n - 1, and the fact that a finite cyclic group of order m contains φ(m) generators.

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