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,and−i, 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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