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

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In summary, the conversation revolved around a discussion of Theorem 8.7 (Fermat's Little Theorem) from Ch. 8 of the book "A First Course in Abstract Algebra" by Anderson and Feil. The questions asked were regarding the proof of the theorem and the explanations for certain statements made in the book. The questions were answered by showing that the statements were either obvious or followed from the definitions and properties of fields.
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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:
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 ...
 
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Peter said:
" ... ... 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 .. ... "
If $[x\cdot i]=[x\cdot j]$, then $x$ can be canceled and we have $i=j$.
 
  • #3
Evgeny.Makarov said:
If $[x\cdot i]=[x\cdot j]$, then $x$ can be canceled and we have $i=j$.
Thanks Evgeny ... grateful for your help ...

Peter
 

1. What is Fermat's Little Theorem?

Fermat's Little Theorem is a fundamental theorem in number theory that states that for any integer a and prime number p, a^p - a is divisible by p. This theorem is named after French mathematician Pierre de Fermat, who first stated it in the 17th century.

2. How is Fermat's Little Theorem used in mathematics?

Fermat's Little Theorem is used in many areas of mathematics, including cryptography, number theory, and algebraic geometry. It is particularly useful in proving the primality of large numbers and in finding inverse elements in modular arithmetic.

3. What is the significance of Anderson and Feil's Theorem 8.7 in relation to Fermat's Little Theorem?

Theorem 8.7 in Anderson and Feil's book "The Theory of Numbers: An Introduction" provides a generalization of Fermat's Little Theorem. It states that for any integer a and positive integer n, a^n - a is divisible by the product of all distinct prime factors of n. This theorem is important in understanding the deeper implications of Fermat's Little Theorem.

4. Can Fermat's Little Theorem be used to prove the primality of all numbers?

No, Fermat's Little Theorem can only be used to prove the primality of certain numbers known as Fermat primes. These are numbers of the form 2^(2^n) + 1, where n is a non-negative integer. However, there are only five known Fermat primes, so Fermat's Little Theorem is not a reliable method for proving primality in general.

5. Are there any other theorems related to Fermat's Little Theorem?

Yes, there are several other theorems that are related to Fermat's Little Theorem, such as Euler's Theorem and Wilson's Theorem. These theorems provide other conditions for divisibility and primality that are based on Fermat's Little Theorem. Additionally, there are many open questions and conjectures related to the generalization of Fermat's Little Theorem, making it a topic of ongoing research in mathematics.

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