Affine Algebraic Curves - Kunz - Exercise 1 - Chapter 1

  • #1
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
 

Answers and Replies

  • #2
mathwonk
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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?
 
  • #3
Math Amateur
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Sorry mathwonk ... not sure ... can you help further ...

Peter
 
  • #4
Math Amateur
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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
 
  • #5
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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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