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

  1. Feb 26, 2017 #1
    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:



    ?temp_hash=bbfb5a4056b517e37a95ed1b249b4210.png
    ?temp_hash=bbfb5a4056b517e37a95ed1b249b4210.png





    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 ...
     

    Attached Files:

  2. jcsd
  3. Feb 26, 2017 #2

    fresh_42

    Staff: Mentor

    It's also obvious. If we assumed ##[x\cdot n]=[x\cdot m]## then what would this mean? You can use all field operations and again the lack of zero divisors. In general, the question to be answered is: why do fields allow multiplicative cancellations? The reason doesn't require this special field, so you may as well prove: ##a\cdot m = b \cdot m \Longrightarrow a = b## for ##a,b \neq 0## of course.
     
  4. Feb 28, 2017 #3
    Thanks for the help fresh_42 ... grateful for your help ...

    If ##[ x \cdot n ] = [ x \cdot m ]## then by cancellation in a field, we have n = m which cannot be the case for the members of the set S ...

    Cancellation works in fields since every element of a field has an inverse ...

    ... so if ##a \cdot m = b \cdot m## then we can post-multiply by ##m^{-1}## and get ##a = b## ...

    Hope that's right ...

    Peter
     
  5. Mar 1, 2017 #4

    fresh_42

    Staff: Mentor

    Well, partially. You cannot rule out the case ##m=0## at prior and operate with ##m^{-1}##.

    Therefore wait as long as you can with additional assumptions.

    This is more of a general advice. Instead operate without (multiplicative) inverse elements.
     
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