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

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In summary: This is always possible, even if there are inverse elements.In summary, fresh_42 is grateful for Peter's help with the proof of Theorem 8.7.
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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:
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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 ...
 

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Math Amateur said:
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 .. ... "
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.
 
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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
 
  • #4
Math Amateur said:
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
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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Related to Fermat's Little Theorem .... Anderson and Feil, Theorem 8.7 .

1. What is Fermat's Little Theorem?

Fermat's Little Theorem is a fundamental theorem in number theory, named after the French mathematician Pierre de Fermat. It states that for any integer a and prime number p, ap - a is divisible by p. In other words, ap is congruent to a modulo p.

2. How is Fermat's Little Theorem useful?

Fermat's Little Theorem has many applications in number theory and cryptography. It is often used in proving the primality of numbers and in checking the primality of large numbers. It is also a key component in many encryption algorithms, such as the RSA algorithm.

3. Can Fermat's Little Theorem be extended to non-prime moduli?

Yes, there is a generalization of Fermat's Little Theorem known as Euler's Theorem, which states that for any integer a and positive integer n, an - a is divisible by n if a and n are coprime (i.e. have no common factors).

4. What are some real-life applications of Fermat's Little Theorem?

Aside from its use in number theory and cryptography, Fermat's Little Theorem has also been applied in fields such as computer science, physics, and economics. For example, it has been used to analyze the running time of certain algorithms, model the dynamics of systems, and understand the behavior of financial markets.

5. Who discovered Fermat's Little Theorem?

Fermat's Little Theorem was first stated by Pierre de Fermat in a letter to mathematician Marin Mersenne in the early 17th century. However, it was not proven until much later by Leonhard Euler in the 18th century. Since then, it has been generalized and extended by many mathematicians, making it one of the most well-known and studied theorems in number theory.

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