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:https://www.physicsforums.com/attachments/4549Hope someone can help ... ...To give a feel for the context and notation I am providing the start to Chapter 1, as follows:View attachment 4550

Peter

 

[jvicens]

, thank you for reaching out for help with Exercise 1 in Chapter 1 of Ernst Kunz's book, "Introduction to Plane Algebraic Curves". I understand how overwhelming complex problems can be, especially when dealing with advanced mathematical concepts.

Before diving into the exercise, let's review the context and notation used in the book. Chapter 1 introduces the basic concepts and definitions of plane algebraic curves. The notation used in the book is standard in algebraic geometry, so it may seem unfamiliar at first. However, with some practice and understanding, it will become more natural.

Now, let's take a closer look at Exercise 1. This exercise is asking you to prove that the set of all points in the plane satisfying the equation x^2 + y^2 = 1 forms a curve. In other words, you need to show that this equation defines a curve in the plane.

To solve this exercise, you will need to use some basic algebraic techniques. First, recall that the equation x^2 + y^2 = 1 represents a circle of radius 1 centered at the origin. This means that all points (x,y) that satisfy this equation will lie on this circle.

Next, you need to show that this circle is a curve. In algebraic geometry, a curve is defined as a set of points that can be described by a single polynomial equation. In this case, the equation x^2 + y^2 = 1 can be written as (x-0)^2 + (y-0)^2 - 1 = 0, which is a polynomial equation in x and y. Therefore, we can conclude that the set of points satisfying this equation forms a curve.

I hope this helps to clarify the context and approach to solving Exercise 1. Don't hesitate to reach out for more assistance if needed. Keep practicing and don't get discouraged, you'll get the hang of it in no time!