How to Interpret and Visualize Affine Algebraic Curves in Ernst Kunz's Book?

[Math Amateur]

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

I need help with interpreting Example 1.2.

The relevant text pertaining to Example 1.2 is as follows:
https://www.physicsforums.com/attachments/4548
Question 1


In Example 1.2 above how do we interpret \(\displaystyle aX + bY + c = 0\)? ... ...I understand that the coefficients come from the field \(\displaystyle K\) ... ... But if we want to plot and examine the curve in the affine plane \(\displaystyle \mathbb{A}^2 (K)\) ... do we take values of \(\displaystyle X, Y\) from \(\displaystyle K\) also? If not what do we do regarding plotting and visualizing the curve in the affine plane ... ?
Question 2In the above text we read the following:" ... ... If \(\displaystyle K_0 \subset K\) is a subfield and \(\displaystyle a,b,c \in K_0\), then the line \(\displaystyle g: aX +bY + c = 0\) certainly possesses \(\displaystyle K_0\)-rational points. ... ... "How do we (formally and rigorously) show that this statement is true?
Hope someone can help ...

Peter
*** NOTE ***

Just a thought on my own Question 2 above ... ...

\(\displaystyle \Gamma \ : \ g = 0\) would contain the points \(\displaystyle X = -c/a, Y= 0\) and \(\displaystyle X = 0, Y = -c/b\) ... that is the points \(\displaystyle ( -c/a, 0), (0, -c/b) \in \Gamma\) ... ...

Further \(\displaystyle K_0\) would contain \(\displaystyle 0\) since \(\displaystyle K_0\) is a subring ... ...So ... ... if we can show that \(\displaystyle K_0\) contains \(\displaystyle -c/a\) and \(\displaystyle -c/b\) we then have shown that the points \(\displaystyle ( -c/a, 0), (0, -c/b) \in K^2\) and hence the points \(\displaystyle ( -c/a, 0), (0, -c/b) \in \Gamma_0 = \Gamma \cap K^2\) ... ... that is that the points \(\displaystyle ( -c/a, 0), (0, -c/b)\) are \(\displaystyle K_0\)-rational points of \(\displaystyle \Gamma\) ... ... BUT \(\displaystyle K_0 \) is a subring not a subfield and so ... we cannot assume \(\displaystyle a^{-1}\) and \(\displaystyle b^{-1} \in K_0\) ...Is my analysis so far correct? ... and if so how do we conclude that the line \(\displaystyle g: aX +bY + c = 0\) possesses \(\displaystyle K_0\)-rational points.
*** NOTE 2 *** (Smirk)

Oh my goodness! Just read Example 1.2 (a) again ... Kunz mentions that he assumes \(\displaystyle K_0 \subset K\) is a subfield ... I was reading the definition of \(\displaystyle K_0\)-rational points of \(\displaystyle \Gamma\) in Definition 1.1 ! ... did not notice that in Example 1.2 (a) he changes the assumption about \(\displaystyle K_0\) ... ...

... hmmm ... must read maths texts more carefully ...

So ... ... now I think the difficulty I mentioned above has gone and the points \(\displaystyle ( -c/a, 0), (0, -c/b)\) are \(\displaystyle K_0\)-rational points of \(\displaystyle \Gamma\) ...

Can someone please confirm that my analysis is correct ... or point out errors ...

 

[jvicens]

Dear Peter,

I am also currently reading Ernst Kunz's book and I would be happy to help you with interpreting Example 1.2.

In Example 1.2, the equation aX + bY + c = 0 represents a line in the affine plane \mathbb{A}^2 (K), where K is the field of coefficients. In order to plot and examine this curve, we can take values for X and Y from the field K. This means that the points on the curve will have coordinates (X, Y) where X and Y are elements of K. For example, if K = \mathbb{Q}, then the points on the curve would have rational coordinates.

As for your second question, it is true that if K_0 \subset K is a subfield and a,b,c \in K_0, then the line g: aX +bY + c = 0 will possess K_0-rational points. This can be shown by considering the points ( -c/a, 0) and (0, -c/b) on the curve. Since K_0 is a subfield, it contains the elements -c/a and -c/b, and therefore these points are K_0-rational. This means that the curve g: aX +bY + c = 0 has at least two K_0-rational points, which satisfies the definition of K_0-rational points in Definition 1.1.

I hope this helps clarify your understanding of Example 1.2. Keep reading and don't hesitate to ask for help if you come across any more difficulties. Good luck!