Kunz - Vanishing Ideal and Minimum Polynomial

[Math Amateur]

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

I need help with some aspects of Kunz' definition of the vanishing ideal of an algebraic curve and Kunz' definition of a minimal polynomial ...

The relevant text from Kunz is as follows:https://www.physicsforums.com/attachments/4552
https://www.physicsforums.com/attachments/4553My questions on the above text are as follows ... ...Question 1In the text above from Kunz, we read the following ...

" ... ... Theorem 1.7 \(\displaystyle \mathscr{J} ( \Gamma )\) is the principal ideal generated by \(\displaystyle f_1 \ ... \ ... \ f_n\). ... ... "

This definition surprised me since I think of a principal ideal as being generated by one element ... ... ? ...

But the I noted that in Definition 1.8 Kunz writes \(\displaystyle \mathscr{J} ( \Gamma ) = (f) \) with \(\displaystyle f \in K [ X,Y ]\) ... ...

SO maybe Kunz is defining \(\displaystyle \mathscr{J} ( \Gamma )\) as a principal ideal generated by \(\displaystyle f\) ... ... but also generated by \(\displaystyle f_1 \ ... \ ... \ f_n\) ... ... is that correct?Can someone please confirm that my analysis is correct ... or possibly correct the errors in my thinking ...
Question 2In the above text, Kunz writes the following:

" ... ... Definition 1.8 Given \(\displaystyle \mathscr{J} ( \Gamma ) = (f)\) with \(\displaystyle f \in K [ X,Y ]\), we call \(\displaystyle f\) a minimal polynomial for \Gamma. ... ... "I am a bit puzzled by this since I am more used to a minimum polynomial being associated with the root \(\displaystyle \alpha\) of a polynomial ... not being associated with an algebraic curve \(\displaystyle \Gamma\) ... ...

... ... indeed a more usual definition is found in Steven Roman's book on Field Theory ... ... as follows:View attachment 4555
https://www.physicsforums.com/attachments/4554

Can someone clarify this by showing that Kunz and Roman's definitions of minimum polynomial are actually the same ... ...

Peter

 

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Question 1In the text above from Kunz, we read the following ...

" ... ... Theorem 1.7 \(\displaystyle \mathscr{J} ( \Gamma )\) is the principal ideal generated by \(\displaystyle f_1 \ ... \ ... \ f_n\). ... ... "

This definition surprised me since I think of a principal ideal as being generated by one element ... ... ? ...

But the I noted that in Definition 1.8 Kunz writes \(\displaystyle \mathscr{J} ( \Gamma ) = (f) \) with \(\displaystyle f \in K [ X,Y ]\) ... ...

SO maybe Kunz is defining \(\displaystyle \mathscr{J} ( \Gamma )\) as a principal ideal generated by \(\displaystyle f\) ... ... but also generated by \(\displaystyle f_1 \ ... \ ... \ f_n\) ... ... is that correct?Can someone please confirm that my analysis is correct ... or possibly correct the errors in my thinking ...

I think you might've misread the notation. He doesn't say that $\mathcal{J}(\Gamma) = (f_1,\ldots, f_h)$ but $\mathcal{J}(\Gamma) = (f_1\cdots f_h)$, in other words, $\mathcal{J}(\Gamma)$ is generated by the product $f_1\ldots f_h$, and not by the individual $f_1,\ldots, f_h$. So he has used the term "principal ideal" in the usual way.

Question 2In the above text, Kunz writes the following:

" ... ... Definition 1.8 Given \(\displaystyle \mathscr{J} ( \Gamma ) = (f)\) with \(\displaystyle f \in K [ X,Y ]\), we call \(\displaystyle f\) a minimal polynomial for \Gamma. ... ... "I am a bit puzzled by this since I am more used to a minimum polynomial being associated with the root \(\displaystyle \alpha\) of a polynomial ... not being associated with an algebraic curve \(\displaystyle \Gamma\) ... ...

... ... indeed a more usual definition is found in Steven Roman's book on Field Theory ... ... as follows:Can someone clarify this by showing that Kunz and Roman's definitions of minimum polynomial are actually the same ... ...

Peter

You haven't made a parallel comparison. Roman is referring to the minimum polynomial for an algebraic element of a field, whereas Kunz is referring to a minimum polynomial for a subset of the affine plane over a field.

 

[Math Amateur]

I think you might've misread the notation. He doesn't say that $\mathcal{J}(\Gamma) = (f_1,\ldots, f_h)$ but $\mathcal{J}(\Gamma) = (f_1\cdots f_h)$, in other words, $\mathcal{J}(\Gamma)$ is generated by the product $f_1\ldots f_h$, and not by the individual $f_1,\ldots, f_h$. So he has used the term "principal ideal" in the usual way.

You haven't made a parallel comparison. Roman is referring to the minimum polynomial for an algebraic element of a field, wheras Kunz is referring to a minimum polynomial for a subset of the affine plane over a field.

Thanks Euge ...

You write:

"I think you might've misread the notation. He doesn't say that $\mathcal{J}(\Gamma) = (f_1,\ldots, f_h)$ but $\mathcal{J}(\Gamma) = (f_1\cdots f_h)$, in other words, $\mathcal{J}(\Gamma)$ is generated by the product $f_1\ldots f_h$, and not by the individual $f_1,\ldots, f_h$. So he has used the term "principal ideal" in the usual way."

Yes, you are quite right ... thanks for that ...
You also write:

"You haven't made a parallel comparison. Roman is referring to the minimum polynomial for an algebraic element of a field, whereas Kunz is referring to a minimum polynomial for a subset of the affine plane over a field."

Yes, OK ... see that ...

Is Kunz definition a common/traditional definition? ... the definitions I found in my textbooks seemed to all refer to an algebraic element of a field ... ...

Peter

 

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Is Kunz definition a common/traditional definition? ... the definitions I found in my textbooks seemed to all refer to an algebraic element of a field ... ...

Yes, I believe Kunz's definition is common.