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I am trying to solve the following problem:

How many solutions does the equation $6x = 5$ have in $\mathbb{Z}/9\mathbb{Z}$

The equivalent of asking:

How many solutions does the equation $6x = 5$ have in $\mathbb{F}9$?

From my understanding $\mathbb{Z}/9\mathbb{Z}$ is simply the set of possible values of integers mod $9$, so it is the set $[0, 9)$ which is equivalent to $\mathbb{F}9$. I just wanted to check that my understanding is correct.

Arturo Magidin
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Thomas Liu
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    If by $\mathbb{F}_9$ you mean the field with nine elements, then no. $\mathbb{Z}/9\mathbb{Z}$ is not isomorphic to the field with nine elements: it's not a field at all. For example, $\overline{3}\neq \overline{0}$, but $\overline{3}\times\overline{3}=\overline{0}$. It's a common beginner's error. – Arturo Magidin Oct 24 '21 at 05:27
  • Your understanding is not correct. Every element of the field $\Bbb F_9$ satisfies $3x=0$. That’s not true of $\Bbb Z/9 \Bbb Z,$ which is not a field. – Robert Shore Oct 24 '21 at 05:27
  • @ArturoMagidin Do you mind providing an example (links, videos, etc.) on how to solve equations in quotient groups? – Thomas Liu Oct 24 '21 at 06:37
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    This is a modular problem, an elementary number theory problem, very little if anything to do with group theory or quotient groups. You are trying to solve the congruence $6x\equiv 5\pmod{9}$. The standard methods for solving this congruence are all you need. – Arturo Magidin Oct 24 '21 at 06:47
  • Your tags were woefully incorrect. What the heck does this have to do with Galois Theory?! – Arturo Magidin Oct 24 '21 at 06:48
  • In a field there is one solution to the equation because each non-zero element of the field has a unique multiplicative inverse. – Mark Bennet Oct 24 '21 at 08:11

1 Answers1

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First of all ,[0,9) represents interval which means it contains all real numbers x such that 0≤x<9.

So, you cannot write it in this way.

More over, Z/9Z is not isomorphic to F9. Z/9Z has zero divisior.(3×3 is equivalent to 0 in Z9) . Remember , Zn is field iff n is prime. so, Z9 cannot be a field.

Nope
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