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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?

Bill Dubuque
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FallInClouds
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community May 26 '23 at 12:48
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    It is easy to see that the different definitions are equivalent, see for example here. A primitive root modulo $p$ is a generator of the multiplicative group of integers modulo $p$. – Dietrich Burde May 26 '23 at 12:52
  • What do you think $\mathbb Z_p$ means, if it isn't defined in terms of congruence? – Thomas Andrews May 26 '23 at 12:54
  • @ThomasAndrews In this book, it said $\mathbb Z_p$ is the set {0,1,...,p-1}. In the definition of primitive root, it just said every nonzero element of $\mathbb Z_p$ is a power of r. But in the example that determine whether 2 and 3 are primitive roots modulo 11, it shows that 2^4=5. I cannot understand it. I think it should be 2^4 mod 11=5. And the definition should be "every nonzero element of $\mathbb Z_p$ is a power of r modulo p" – FallInClouds May 26 '23 at 17:24
  • In $\mathbb Z_p,$ multiplication is redefined as $a\cdot_p b= (ab)\bmod p.$ So $2^4=5,$ not $2^4\bmod 11=5.$ The modulo 11 is already taken into account by the system's definition of multiplication. – Thomas Andrews May 26 '23 at 18:47
  • @ThomasAndrews It explains the matter! – FallInClouds May 27 '23 at 04:45
  • See the linked dupe for more on the meaning of $\Bbb Z_n$ and its relationship with congruences $!\bmod n\ \ $ – Bill Dubuque May 27 '23 at 14:53

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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.

Anne Bauval
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  • Both definitions are usual. Btw, in French we have two words: "inversible" for your multiplicative inverses, and "symétrisable" for your general inverses (for any binary operation, be it multiplication or addition or whatsoever). – Anne Bauval May 26 '23 at 17:47