Part 1: Algebraic Preliminaries
Chapter 6: Equations and varieties
Logic of Equality
An equation is a statement, or assertion, that one thing is identical to another. In mathematics we replace is by = and use symbols that stand for the terms.
History of equations
Long before algebra as we know it, ancient peoples were working with equations.
A triangle whose three sides have lengths 3, 4, and 5 is a right triangle which is an example of a Diophantic equation because the unknowns are restricted to integers. Around the late 1500s Descartes added the connection between algebra and geometry now known as analytic geometry. Descartes, as a philosopher believed that the physical universe was governed entirely by the laws of geometry. Newton ( and Leibniz ) discovered that this wasn't true, they had to invent calculus to solve their scientific problems mathematically.
Z-Equations
A rational number is any number that can be expressed as the ratio of two integers. Real numbers are rational iff it is a terminating decimal or a repeating decimal. The set of all rational numbers is usually denoted as Q. We will deal mostly with equations where all the constants are integers. Or equations "defined over the integers". A Z-Equation is an equality of polynomials with integer coefficients. One of the main problems in number theory is finding and understanding all solutions of Z-equations.
Varieties
Fix the attention on a particular Z-equation. Write S(Z) for the set of all integral solutions of that equation, S(Q) for the set of all rational solutions of it, and so on. We call S an "algebraic variety". The variety S defined by a Z-equation ( or a system of Z-equations ) is the function that assigns to any number system the set of solutions S(A) of the equation or system of the equations.
For example define the Variety S as x^2 + Y^2 = 1
Then
S(Z) = {{1,0),(0,1),(-1,0),(0,-1)}.
S(Q) = {t in Q | 1-t^2 / 1+t^2, 2t/1+t^2}.
We can reformulate Fermat's Last Theorem using varieties as follows.
For any positive integer n, let V_n be the variety defined by x^n + y_n = z^n. Then if n > 2, V_n(Z) contains only solutions where one or more of the variables is 0.
Systems of equations
The system
x^2+y^2=1
x>0
is valid and has solutions, but it does NOT define an algebraic variety because inequalities are not defined in C nor in any of the finite fields.
Take the system
x^2+y^2+z^2=w
w^4=1
x+y=z
then S(R) is the ellipse x^2+y^2+x y = 1/2.
Finding roots of polynomials
The easiest general class of varieties to look at would be those defined by a single Z-equation in a single variable, for instance, x^3 + x - 2 = 0. The study of this type of variety is dominated by the concept of the Galois group. ( More in 8 and 13 ). If f(x) is a polynomial the roots of f(x) are the numbers c such that f(c)=0.
Are There General Methods for Finding Solutions to Systems of Polynomial Equations?
On a purely number-theoretical level, leaving philosophy and logic behind, we also have the famous theorem of Abel and Ruffini: Unlike quadratic polynomials, for which we can use the quadratic formula, for polynomials f (x) of degree 5 or greater, there is no formula involving just addition, subtraction, multiplication, division, and nth roots (n = 2, 3, 4, . . .) that can solve f (x) = 0 in general.
Deeper understanding is desirable
The amazing discovery of Galois is that there is more structure to S(A). As we shall see, S(A) is not just a set; it is the basis for defining a representation of a certain group, called the Galois group. We will look at another series of very interesting and very important, though not so very simple, Z-varieties: elliptic curves. These two kinds of varieties will give us some of our main examples to help us understand Galois groups and their representations.
To be continued with 7. Quadratic reciprocity
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