Existence and Uniqeness of Finite Fields ....

In summary, a finite field is a mathematical structure with a finite set of elements and two operations, addition and multiplication, that satisfies certain properties. It must contain a prime number of elements and its existence and uniqueness have important implications in various areas of mathematics and computer science. The existence and uniqueness of finite fields can be proven using the existence and uniqueness theorems. Some real-life applications of finite fields include error-correcting codes, encryption algorithms, and generating random numbers.
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I am reading David S. Dummit and Richard M. Foote : Abstract Algebra ...

I am trying to understand the example on Finite Fields in Section 13.5 Separable and Inseparable Extensions ...The example reads as follows:
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My questions are as follows:
Question 1In the above text from D&F we read the following:

" ... ... If ##\mathbb{F}## is of dimension ##n## over its prime subfield ##\mathbb{F}_p##, then ##\mathbb{F}## has precisely ##p^n## elements. ... ... "Can someone please explain why, exactly, this follows?

Question 2

In the above text from D&F we read the following:

" ... ... Since the multiplicative group ##\mathbb{F}^{ \times }## is (in fact cyclic) of order ##p^n - 1##, we have ##\alpha^{ p^n - 1 } = 1## for every ##\alpha \ne 0 in \mathbb{F}## ... ... "Can someone give me the exact reasoning concerning why ##\mathbb{F}^{ \times }## being of order ##p^n - 1## implies that ##\alpha^{ p^n - 1} = 1## for every ##\alpha \ne 0## in ##\mathbb{F}## ... ... ?(I am guessing that for some reason I cannot explain, that ##\mathbb{F}^{ \times }## being of order ##p^n - 1 ## implies that the characteristic is ##p^n - 1## ... ... but why does it mean this is the case ...? )

Hope someone can help ...

Peter
 

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  • #2
Math Amateur said:
Question 1
In the above text from D&F we read the following:
" ... ... If ##\mathbb{F}## is of dimension ##n## over its prime subfield ##\mathbb{F}_p##, then ##\mathbb{F}## has precisely ##p^n## elements. ... ... "
What does it mean "of dimension ##n##"? If you write down all possible linear combinations of basis vectors, how many possibilities do you get?
Question 2
In the above text from D&F we read the following:
" ... ... Since the multiplicative group ##\mathbb{F}^{ \times }## is (in fact cyclic) of order ##p^n - 1##, we have ##\alpha^{ p^n - 1 } = 1## for every ##\alpha \ne 0 in \mathbb{F}## ... ... "

Can someone give me the exact reasoning concerning why ##\mathbb{F}^{ \times }## being of order ##p^n - 1## implies that ##\alpha^{ p^n - 1} = 1## for every ##\alpha \ne 0## in ##\mathbb{F}## ... ... ?
If you have any finite group ##G##, say ##|G|=m## and take an element ##g \in G##, then consider the subgroup ##U:=\langle g \rangle \subseteq G##. What can be said about the order of ##U## and therewith the order of ##g##? Now with that, what do you get for ##g^m##?
(I am guessing that for some reason I cannot explain, that ##\mathbb{F}^{ \times }## being of order ##p^n - 1 ## implies that the characteristic is ##p^n - 1## ... ... but why does it mean this is the case ...? )
No. The characteristic of a field always has to be either ##0## or prime, and ##p^n-1## is only in one single case prime whereas the statement is generally valid. The characteristic is the smallest number ##n## such that ## \underbrace{1+ \ldots + 1}_{n-times} =0## or zero, if ##n## isn't finite. The order of ##\mathbb{F}^\times## is ##m=|\mathbb{F}|-|\{0\}|=p^n-1## not the characteristic.
 
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  • #3
fresh_42 said:
What does it mean "of dimension ##n##"? If you write down all possible linear combinations of basis vectors, how many possibilities do you get?

If you have any finite group ##G##, say ##|G|=m## and take an element ##g \in G##, then consider the subgroup ##U:=\langle g \rangle \subseteq G##. What can be said about the order of ##U## and therewith the order of ##g##? Now with that, what do you get for ##g^m##?

No. The characteristic of a field always has to be either ##0## or prime, and ##p^n-1## is only in one single case prime whereas the statement is generally valid. The characteristic is the smallest number ##n## such that ## \underbrace{1+ \ldots + 1}_{n-times} =0## or zero, if ##n## isn't finite. The order of ##\mathbb{F}^\times## is ##m=|\mathbb{F}|-|\{0\}|=p^n-1## not the characteristic.
Hi fresh_42 ... ... thanks for the help ...

Regarding Question 1, you write the following"

"What does it mean "of dimension n" role="presentation">n"? If you write down all possible linear combinations of basis vectors, how many possibilities do you get?"

Well, dimension ##n## means there are ##n## basis vectors, and each linear combination of the basis vectors has ##n## coefficients with ##p## possibilities for each ... ... so we get a total of ##p^n## unique elements in the field ##\mathbb{F}## ... ... Is that the correct analysis/reasoning?

Still thinking about Question 2 ...

Peter
 
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  • #4
Math Amateur said:
Hi fresh_42 ... ... thanks for the help ...

Regarding Question 1, you write the following"

"What does it mean "of dimension n" role="presentation">n"? If you write down all possible linear combinations of basis vectors, how many possibilities do you get?"

Well, dimension ##n## means there are ##n## basis vectors, and each linear combination of the basis vectors has ##n## coefficients with ##p## possibilities for each ... ... so we get a total of ##p^n## unique elements in the field ##\mathbb{F}## ... ... Is that the correct analysis/reasoning?

Still thinking about Question 2 ...

Peter
Yes. And they are all different, because ##\alpha_1 b_1 + \ldots + \alpha_n b_n = \beta_1 b_1 + \ldots + \beta_n b_n## means, that each ##\alpha_i=\beta_i## if ##\{ b_1,\ldots ,b_n\}## is a basis. So there cannot be two different combinations resulting in the same vector, i.e. all ##p^n## combinations are different elements.
 
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FAQ: Existence and Uniqeness of Finite Fields ....

1. What is the definition of a finite field?

A finite field is a mathematical structure that consists of a finite set of elements and two operations, addition and multiplication, which satisfy certain properties.

2. How many elements does a finite field contain?

A finite field must contain a prime number of elements, denoted by p. The number of elements in a finite field is given by p^n, where n is a positive integer.

3. What is the significance of the existence and uniqueness of finite fields?

The existence and uniqueness of finite fields have important implications in various areas of mathematics and computer science, including cryptography, coding theory, and algebraic geometry.

4. How can the existence and uniqueness of finite fields be proven?

The existence of finite fields can be proven using the existence theorem, which states that for every prime number p and positive integer n, there exists a finite field with p^n elements. The uniqueness of finite fields can be proven using the uniqueness theorem, which states that any two finite fields with the same number of elements are isomorphic.

5. What are some real-life applications of finite fields?

Finite fields have numerous practical applications, including error-correcting codes used in communication systems, encryption algorithms in cryptography, and generating random numbers in computer simulations and games.

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