Affine Algebraic Curves - Kunz - Exercise 1 - Chapter 1

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SUMMARY

The discussion revolves around Exercise 1 from Ernst Kunz's "Introduction to Plane Algebraic Curves," focusing on the properties of polynomials over fields. Specifically, the question posed is whether the last root of a polynomial of degree d, with d-1 roots in field F, must also reside in F. Participants clarify that if F is an algebraically closed field, then all d roots are indeed in F. Furthermore, the distinction between fields F and F0 is emphasized, highlighting the importance of understanding the implications of algebraically closed fields in polynomial root behavior.

PREREQUISITES
  • Understanding of polynomial functions and their degrees
  • Knowledge of algebraically closed fields
  • Familiarity with the concept of roots of polynomials
  • Basic principles of field theory
NEXT STEPS
  • Study the properties of algebraically closed fields in detail
  • Explore the implications of the Fundamental Theorem of Algebra
  • Learn about polynomial factorization in different fields
  • Investigate the relationship between roots and coefficients in polynomials
USEFUL FOR

Mathematicians, students of algebra, and anyone interested in the properties of polynomial equations and field theory will benefit from this discussion.

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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