Affine Algebraic Curves - Kunz - Exercise 1 - Chapter 1

[Math Amateur]

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:

Hope someone can help ... ...To give a feel for the context and notation I am providing the start to Chapter 1, as follows:

Peter

 

[mathwonk]

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?

 

[Math Amateur]

Sorry mathwonk ... not sure ... can you help further ...

Peter

 

[Math Amateur]

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

 

[micromass]

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 $\beta\in F_0$?