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No credit card required 58 CHAPTER 6
There are even more C-solutions. First, from (2) we see that w
can be any one of the four fourth roots of 1.
3
The same reasoning
as before tells us that for each choice of a fourth root of 1 to be the
value of w, we will get a n inﬁnite number of possibilities for (x, y, z).
It is easy to write down systems or even single equations whose
solution set is very hard to determine, or even unknown.
EXERCISE: Let S be the variety deﬁned by
1. x
17
+ 33x
2
y
3
xyz = 44.
2. x
3
+ y
3
= z
2
+ 137.
What is S(Z)?
SOLUTION: We have no idea—it is probably very hard to
ﬁgure this out. Even if it turns out that by some trick you can
solve this system of equations (maybe reducing modulo a
prime could show there are no integer solutions), just replace
it by an even more complicated system of your invention.
In summary, given a system of Z-equations, we can consider the
variety S of simultaneous solutions. Then for any number system
A,wegetthesetS(A) of simultaneous solutions where the values
of the variables are drawn from the system A.Wewillseemany
more examples of varieties as we go along. We will try to get a feel
for some varieties that are particularly interesting because of their
structure.
Equivalent Descriptions of the Same Variety
Many different systems of equations can deﬁne the same variety
S. It is important to understand this concept, so we give a few
examples. For instance, any set of inconsistent equations will deﬁne
the empty variety.
3
The 4 fourth roots of 1 are 1, 1, i,andi, as you can see by raising each of them to the
fourth power. It can be proven that every nonzero complex number has n different nth
complex roots for all n 1.
EQUATIONS AND VARIETIES 59
EXAMPLE: If S is the variety deﬁned by the system:
1. x = y
2. x = y + 1
then S(A) is the empty set for any number system A.If
instead we consider the system
1. x = y
2
2. x = y
2
1
and let T be the variety it deﬁnes, then T is also the empty
variety. So S = T even though the systems of equations are
different.
EXAMPLE: Consider the two systems
1. 2x + 3y = 7.
2. x y = 6.
and
1
. 3x + 2y = 13.
2
. 5x + 5y = 20.
We get (1
) by adding (1) and (2). We get (2
) by doubling (1)
and adding it to (2). Similarly, we can get the ﬁrst system of
equations from the second by
subtracting (1
)from(2
) to get (1);
doubling (1
) and subtracting (2
) to get (2).
Therefore, the two systems are totally equivalent: they have
the same solution sets in any number system. They deﬁne the
same variety.
EXAMPLE: Yet another:
1. x
2
+ y
3
= 0.
1
. (x
2
+ y
3
)
2
= 0.
Because only 0
2
is 0, these two equations deﬁne the same
variety, whatever it is.

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