Answered: What are all of the distinct generators…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.1: Definition Of A Ring

Problem 44E: 44. Consider the set of all matrices of the form, where and are real numbers, with the...

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What are all of the distinct generators of Z₁₇with respect to multiplication?
What are all the distance generators of Z₁₉with respect to multiplication?

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We have to find the all distinct generators of $Z_{17}$ with respect to multiplication.

Recall the corollary

The multiplication group of all nonzero elements of a finite field is cyclic.

Note that the unity of $Z_{17}$ is $1$ .

If $a$ is a generator of the multiplicative group ${Z^{*}}_{17}$ if and only $a^{n} \neq 1$ , where $n$ goes over divisors of $16$ .

Compute $3^{8}$ in $Z_{17}$

$\begin{matrix} 3^{8} = 6561 \\ = 6545 + 6 \\ = 385 \times 17 + 6 \\ \equiv 6 (\mod 17) \end{matrix}$

Since, $3^{8} \equiv 1 (\mod 17)$

Hence, $3$ is a generator of ${Z^{*}}_{17}$ .

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