Affine Algebraic Curves - Kunz - Exercise 1 - Chapter 1

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Discussion Overview

The discussion revolves around Exercise 1 from Chapter 1 of Ernst Kunz's book "Introduction to Plane Algebraic Curves." Participants are exploring the implications of polynomial roots in relation to fields, particularly focusing on the conditions under which all roots of a polynomial of degree d are contained within a given field F.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant questions whether having d-1 roots in a field F allows for the conclusion that the last root is also in F.
  • Another participant suggests that if F is an algebraically closed field, then all d roots of the polynomial would indeed be in F.
  • A later reply emphasizes the distinction between being in field F and a subfield F_0, prompting further consideration of the implications of roots in relation to these fields.
  • There is a hint provided regarding the polynomial form (X-α)(X-β) and the relationship between α and β concerning their membership in F_0.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the conditions under which the last root can be concluded to be in F. Multiple competing views remain about the implications of algebraically closed fields and the specific fields being referenced.

Contextual Notes

There is a lack of consensus on the definitions of the fields involved and the assumptions regarding the nature of the polynomial and its roots. The discussion does not resolve the mathematical steps necessary to reach a conclusion.

Math Amateur
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I am reading Ernst Kunz book, "Introduction to Plane Algebraic Curves"

I need help with Exercise 1, Chapter 1 ...

Indeed ... I am a bit overwhelmed by this problem ..

Exercise 1 reads as follows:

Kunz - Exercise 1 - Chapter 1.png
Hope someone can help ... ...To give a feel for the context and notation I am providing the start to Chapter 1, as follows:
Kunz - Exercise 1 - Chapter 1.png
Kunz - Some Basic Definitions - Chapter 1.png
Kunz - Some Basic Definitions - Chapter 1.png

Peter
 
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suppose you have a polynomial of degree d over F and d-1 roots are in F. Can you conclude that the last root is also in F?
 
Sorry mathwonk ... not sure ... can you help further ...

Peter
 
Well ... maybe if F is an algebraically closed field, then all the d roots of the polynomial would be in F ...

Is that correct?

Peter
 
Math Amateur said:
Well ... maybe if F is an algebraically closed field, then all the d roots of the polynomial would be in F ...

Is that correct?

Peter

You don't want them to be in ##F##, you want them to be in ##F_0##. Think about mathwonk's hint. If ##(X-\alpha)(X-\beta)## is a polynomial in ##F_0## such that ##\alpha\in F_0##, why is ##\beta\in F_0##?
 

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