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ELLIPTIC CURVES 109
The particular details of the abelian group law actually do not
concern us very much. What does matter is that if we combine two
points in E(Q), for example, then the result is also in E(Q). This
is not true for E(Z) because the number λ might be a fraction, and
then we might get fractional values for x
3
and y
3
. But it is true
for any number system in which we can divide. So if we start with
elements of E(R), and add them, we will get elements of E(R); ditto
for E(C) and E(F
p
).
Digression: What Is So Great about Elliptic Curves?
The question should be, “What is not great about them?” They were
essential in Wiles’s proof of Fermat’s Last Theorem. You can read
a brief description in our paper (Ash and Gross, 2000), or look at
some of the many books on the subject (Hellegouarch, 2002; Singh,
1997; van der Poorten, 1996), or skip ahead to chapter 22. Elliptic
curves have given rise to a tremendous amount of interesting
number theory, including the part we are trying to explain in this
book. Mathematicians are just lucky that elliptic curves are fairly
simple—they only involve two variables and no powers higher than
the cube—and yet are so rich.
One can see how people began to be curious about elliptic
curves—even ancient Greeks such as Diophantus, although he did
not call them elliptic curves. As we have seen, equations in only
one variable can be difﬁcult to solve; even formulas for the cubic
and quartic equations were not discovered until the Renaissance.
But we go on to one equation in two variables anyway. If all
the exponents are 1 or less, the solution set can be found by
simple arithmetic. Uninteresting. (See chapter 10 about how you
can use matrices to solve such equations.) If all the exponents
are 2 or less, the solution set can be understood in terms of
what are called “conic sections, and some of the later Greeks
understood those very well, although it was a fairly advanced
topic for them. So the natural thing to do, if you are a mathe-
matician, is to keep going until you get stuck: Throw in a cubic

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