Prime Subfiellds - Lovett, Proposition 7.1.3 ....

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Discussion Overview

The discussion revolves around Proposition 7.1.3 from Stephen Lovett's "Abstract Algebra: Structures and Applications," specifically focusing on the nature of the subfield defined by the elements of the finite field $\mathbb{F}_p = \mathbb{Z} / p \mathbb{Z}$ for a prime $p$. Participants explore the implications of this proposition and seek clarification on the structure and properties of finite fields.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Peter questions how the subfield containing elements from 0 to \( p-1 \) can be equal to \( \mathbb{Z} / p \mathbb{Z} \) given that it includes divisions of these elements.
  • Evgeny points out that \( \mathbb{Z} / p \mathbb{Z} \) is a field for a prime \( p \) and thus contains multiplicative inverses for all nonzero elements.
  • Peter acknowledges his uncertainty regarding the properties of finite fields and expresses a need to read more on the topic.
  • Another participant suggests Peter revisit his earlier posts on finite fields for additional context.
  • Peter reflects on his previous engagement with finite fields and recognizes a lack of depth in his understanding.
  • HallsofIvy provides an example using \( p = 5 \) to illustrate how multiplication modulo \( p \) works and how multiplicative inverses are derived within the field.
  • Peter expresses gratitude for the helpfulness of HallsofIvy's explanation.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the initial confusion regarding the equivalence of the subfield and \( \mathbb{Z} / p \mathbb{Z} \). There are multiple viewpoints on the understanding of finite fields, with some participants providing clarifications while others express uncertainty.

Contextual Notes

Peter's understanding of finite fields appears to be limited, and there are references to earlier posts that may contain relevant information. The discussion includes assumptions about the properties of fields that may not be fully articulated by all participants.

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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 $$\mathbb{F}_p = \mathbb{Z} / p \mathbb{Z}$$. ... ... " ... ... BUT ... the subfield contains elements $$0, 1, 2, 3, 4, 5, \ ... \ ... \ (p -1)$$ ... and being a field, it contains divisions of these elements such as $$1/2, 3/5 \ ... \ ... \ ...$$... so how can this subfield be equal to $$\mathbb{Z} / p \mathbb{Z}$$ ... ... ?
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?
 
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
 
Hi Peter,

I would suggest looking back at your earlier posts on finite fields, in particular, one of your featured posts on finite fields.
 
For example, if p= 5 them F_p 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".
 
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

- - - Updated - - -

HallsofIvy said:
For example, if p= 5 them F_p 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
 

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