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Affine Algebraic Curves - Kunz - Exercise 1 - Chapter 1

  1. Jul 19, 2015 #1
    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
     
  2. jcsd
  3. Jul 19, 2015 #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?
     
  4. Jul 19, 2015 #3
    Sorry mathwonk ... not sure ... can you help further ...

    Peter
     
  5. Jul 19, 2015 #4
    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
     
  6. Jul 20, 2015 #5
    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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