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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 \(\displaystyle S\) 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 \(\displaystyle S\) 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 ...