Exercise 1840Number theory › PrimesL1Added 3 Nov 2026

Finding primitive roots

#primitive root#mod prime#computation

Statement

Find a primitive root mod $7$ .

Hint

The group $\mathbb{F}_7^*$ has order $6$ ; test elements $2,3,\ldots$ until you find one of order $6$ .

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Test $2$ . Powers: $2,4,1,2,\ldots$ : order $3$ , not primitive.

Test $3$ . Powers: $3,9\equiv2,6,4,5,1$ : order $6$ , primitive.

Conclusion. $3$ is a primitive root mod $7$ ; every element of $\mathbb{F}_7^*$ is a power of $3$ .