I've some problem in the definition of primitive root in the Discrete Mathematics and Its Applications

Question

In the book, he said that "A primitive root modulo a prime p is an integer r in $\mathbb Z_p$ such that every nonzero element of $\mathbb Z_p$ is a power of r."
It is very different to other definition of primitive root on the web, in which they all mentioned cogruence. In the book I mentioned, there is an example that determine whether 2 and 3 are primitive roots modulo 11. It shows that 2⁴=5. I think it is wrong, it should be 2⁴ mod 11=5. And the definition should be "A primitive root modulo a prime p is an integer r in $\mathbb Z_p$ such that every nonzero element of $\mathbb Z_p$ is a power of r modulo p."
How do you think about it?

Answer 1

Your perplexity is legitimate, because the book mixes two common (very near, though strictly speaking non equivalent) definitions:

A primitive root modulo $n$ is:

• an element $a$ of $\Bbb Z_n$ such that every invertible element of $\mathbb Z_n$ is a power of a.
• an integer $r$ such that every integer coprime to $n$ is congruent $\bmod n$ to a power of $r.$

The integers $r$ of the second definition are the elements of the congruence classes $a$ of the first one.