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Fearless Symmetry by Robert Gross, Avner Ash

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COMPLEX NUMBERS 47
Complex Numbers and Solving Equations
What is the point? First, mathematicians proved that using the
set of complex numbers, which we call C, will not lead to any
contradictions. Even though you might not like the idea of a square
root of 1, pretending that it exists will not get you into logical
trouble.
Second, and this is really the key point, the job of adding on
square roots (or cube roots, or anything else that you might like
to add on) stops with C. You do not need to add on some other new
symbol j with j
2
= i, for example; there already is a square root of i
in C.
EXERCISE: Let α =
1
2
+
1
2
i
.Checkthatα
2
= i.
In fact, every root of a complex number is already in C. But even
more than that is true:
THEOREM 5.2: Let f (x) be a polynomial whose coefficients
are any complex numbers. (For example, f (x) might have
integer coefficients .) Then the equation f (x) = 0 has solutions
in C.
Digression: Theorem
A theorem is a mathematical statement that can be proved to be
true. Despite the similarity to the word “theory, there is nothing
speculative about a theorem.
Modern mathematical usage distinguishes between various
terms, all of which refer to mathematical statements that are
proved. The word lemma refers to a minor “helping” statement
used along the way toward proving a theorem. A corollary is an
immediate and easy consequence of a theorem.
Algebraic Closure
The fancy way of conveying the content of Theorem 5.2 is to say
that C is algebraically closed. The first really good proof was given
48 CHAPTER 5
by Gauss, though earlier mathematicians such as Euler knew this
fact as well.
5
We are not going to need to use all of C very often. It contains
numbers such as π, which cannot be understood algebraically. W e
will restrict ourselves mostly to that part of C which consists of
the solutions of all equations f (x) = 0wheref (x) is a polynomial
with integer coefficients. That set is called Q
alg
, and it exists as a
subset of C.Infact,Q
alg
itself is a field! Sample elements of Q
alg
:
0, 1, 3,
1
4
,
11, i,
3
2 3i, and the solutions in C of the equation
x
5
+ x 1 = 0.
5
Consult (Nahin, 1998) for a historical summary.

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