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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 $$\mathscr{J} ( \Gamma )$$ is the principal ideal generated by $$f_1 \ ... \ ... \ f_n$$. ... ... "
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 $$\mathscr{J} ( \Gamma ) = (f) $$ with $$f \in K [ X,Y ]$$ ... ...
SO maybe Kunz is defining $$\mathscr{J} ( \Gamma )$$ as a principal ideal generated by $$f$$ ... ... but also generated by $$f_1 \ ... \ ... \ f_n$$ ... ... 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 $$\mathscr{J} ( \Gamma ) = (f)$$ with $$f \in K [ X,Y ]$$, we call $$f$$ 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 $$\alpha$$ of a polynomial ... not being associated with an algebraic curve $$\Gamma$$ ... ...
... ... 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
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 $$\mathscr{J} ( \Gamma )$$ is the principal ideal generated by $$f_1 \ ... \ ... \ f_n$$. ... ... "
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 $$\mathscr{J} ( \Gamma ) = (f) $$ with $$f \in K [ X,Y ]$$ ... ...
SO maybe Kunz is defining $$\mathscr{J} ( \Gamma )$$ as a principal ideal generated by $$f$$ ... ... but also generated by $$f_1 \ ... \ ... \ f_n$$ ... ... 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 $$\mathscr{J} ( \Gamma ) = (f)$$ with $$f \in K [ X,Y ]$$, we call $$f$$ 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 $$\alpha$$ of a polynomial ... not being associated with an algebraic curve $$\Gamma$$ ... ...
... ... 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