Fermat's Little Theorem .... Anderson and Feil, Theorem 8.7 .... ....

[Math Amateur]

I am reading Anderson and Feil - A First Course in Abstract Algebra.

I am currently focused on Ch. 8: Integral Domains and Fields ...

I need some help with an aspect of the proof of Theorem 8.7 (Fermat's Little Theorem) ...

Theorem 8.7 and its proof read as follows:
View attachment 6435
https://www.physicsforums.com/attachments/6436

My questions regarding the above are as follows:
Question 1

In the above text from Anderson and Feil we read the following:

" ... ... Because a field has no zero divisors, each element of $$S$$ is non-zero ... "Can someone please demonstrate exactly why this follows ... ?

Question 2

In the above text from Anderson and Feil we read the following:" ... ... Because a field satisfies multiplicative cancellation, no two of these elements are the same .. ... "Can someone please demonstrate exactly why it follows that no two of the elements of $$S$$ are the same .. ... "
Help will be appreciated ...

Peter*** EDIT ***
oh dear ... can see that the answer to Question 1 is obvious ... indeed it follows from the definition of zero divisor ... apologies ... brain not in gear ...

 

[Evgeny.Makarov]

" ... ... Because a field satisfies multiplicative cancellation, no two of these elements are the same .. ... "Can someone please demonstrate exactly why it follows that no two of the elements of $$S$$ are the same .. ... "

If $[x\cdot i]=[x\cdot j]$, then $x$ can be canceled and we have $i=j$.

 

[Math Amateur]

If $[x\cdot i]=[x\cdot j]$, then $x$ can be canceled and we have $i=j$.

Thanks Evgeny ... grateful for your help ...

Peter