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Abstract Algebra

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Primitive Root mod p

ABSALG-AY4TTU

Let $p\ge 2$ be prime.

An integer $a$ is called a primitive root mod $p$ if there is no positive integer $h$ strictly smaller than $p-1$ with $a^h\equiv 1$ mod $p$.

Which of the following is true?

A

There are no primitive roots mod $3$.

B

If $a\not\equiv0$ mod $p$ and $h$ is the smallest positive integer with $a^h\equiv 1$ mod $p$, then $h$ divides $p-1$.

C

There are $3$ primitive roots mod $5$.

D

$3$ is not a primitive root mod $7$.

E

If $a$ is a primitive root mod $p$, then so is $a^k$, for all $k=1,\ldots,p-1$.