Prime Subfiellds - Lovett, Proposition 7.1.3 ....

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In summary, Chapter 7 of the book "Abstract Algebra: Structures and Applications" by Stephen Lovett discusses the field extension of a prime number field. Proposition 7.1.3 plus some introductory remarks (proof?) mentions that the subfield of a field containing all nonzero elements has the structure of $\mathbb{F}_p = \mathbb{Z} / p \mathbb{Z}. However, the multiplication on these elements as defined by distributivity gives this set of elements the structure of \mathbb{F}_p = \mathbb{Z} / p \mathbb{Z}. This is due to the presence of multiplicative inverses of all nonzero elements in
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I am reading Abstract Algebra: Structures and Applications" by Stephen Lovett ...

I am currently focused on Chapter 7: Field Extensions ... ...

I need help with the proof of, or at least some remarks concerning, Proposition 7.1.3 ...Proposition 7.1.3 plus some introductory remarks (proof?) reads as follows:
https://www.physicsforums.com/attachments/6543

In the above text from Lovett we read the following:"... ... However, the multiplication on these elements as defined by distributivity gives this set of elements the structure of \(\displaystyle \mathbb{F}_p = \mathbb{Z} / p \mathbb{Z}\). ... ... " ... ... BUT ... the subfield contains elements \(\displaystyle 0, 1, 2, 3, 4, 5, \ ... \ ... \ (p -1)\) ... and being a field, it contains divisions of these elements such as \(\displaystyle 1/2, 3/5 \ ... \ ... \ ...\)... so how can this subfield be equal to \(\displaystyle \mathbb{Z} / p \mathbb{Z}\) ... ... ?
Hope someone can help ...

Peter
 
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Are you aware that $\mathbb{Z} / p \mathbb{Z}$ is a field for a prime $p$ and therefore contains multiplicative inverses of all nonzero elements?
 
  • #3
Evgeny.Makarov said:
Are you aware that $\mathbb{Z} / p \mathbb{Z}$ is a field for a prime $p$ and therefore contains multiplicative inverses of all nonzero elements?
Thanks Evgeny ... hmmm ... certainly seems that Lovett assumed I was aware of that ... ...

I was dimly aware ... but uncertain of how and why it all worked out ...

Intend to read material on finite fields in some of my texts ...

Peter
 
  • #4
Hi Peter,

I would suggest looking back at your earlier posts on finite fields, in particular, one of your featured posts on finite fields.
 
  • #5
For example, if p= 5 them [tex]F_p[/tex] has the set {0, 1, 2, 3, 4} with "multiplication" being "multiplication modulo 5". In particular, 2*3= 6= 1 (mod 5) so "1/2" is "3" and "1/3" is "2". 4*4= 16= 1 (mod 5) so 4 is its own multiplicative inverse: "1/4" is "4".
 
  • #6
Euge said:
Hi Peter,

I would suggest looking back at your earlier posts on finite fields, in particular, one of your featured posts on finite fields.
Yes ... good suggestion Euge ...

On my excursion into finite fields, I obviously did not spend enough time and effort on the topic ...

Peter

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HallsofIvy said:
For example, if p= 5 them [tex]F_p[/tex] has the set {0, 1, 2, 3, 4} with "multiplication" being "multiplication modulo 5". In particular, 2*3= 6= 1 (mod 5) so "1/2" is "3" and "1/3" is "2". 4*4= 16= 1 (mod 5) so 4 is its own multiplicative inverse: "1/4" is "4".
Thanks HallsofIvy ... your post was most helpful ...

Peter
 

FAQ: Prime Subfiellds - Lovett, Proposition 7.1.3 ....

What is Proposition 7.1.3 in Lovett's Prime Subfields?

Proposition 7.1.3 is a mathematical statement in Lovett's Prime Subfields that states that if a finite field has a prime number of elements, then it is a prime subfield of itself.

Why is Proposition 7.1.3 important?

Proposition 7.1.3 is important because it provides a key property of prime subfields in finite fields. It shows that every finite field with a prime number of elements can be considered as a prime subfield of itself, which is a fundamental concept in abstract algebra.

How is Proposition 7.1.3 proved?

Proposition 7.1.3 is proved using basic properties of finite fields and the definition of prime subfields. It can be shown through a direct proof or by contradiction.

What implications does Proposition 7.1.3 have in other areas of mathematics?

Proposition 7.1.3 has implications in various areas of mathematics, such as algebraic geometry, coding theory, and cryptography. It helps to establish the structure and properties of finite fields, which are important in these fields.

Are there any real-world applications of Proposition 7.1.3?

Yes, there are many real-world applications of Proposition 7.1.3, particularly in the fields of coding theory and cryptography. It is used to construct error-correcting codes and secure communication systems, which are essential in modern technology.

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