The
set of
complex numbers - denoted by
C - is the
algebraic closure of the
real numbers, i.e. the smallest
field contaning the reals such that every
polynomial equation is solvable. This is probably the most important property of the complex numbers.
This does not imply that it is convenient to define C as the algebraic closure of the reals. The following definition is somewhat easier to work with.
Definition:
C is the smallest field (up to an isomorphism) containing R as a subfield such that the equation X2 + 1 = 0 has a solution in C.
Proof of existence:
Let C0 be R2 together with addition and multiplication defined in the following way:
(a, b) + (c, d) = (a+c, b+d)
(a, b) * (c, d) = (ac - bd, ad + bc)
It is easily verified that C0 has all the required properties of a field. The subfield R x {0} of C0 is isomorphic to R, and the element i = (0, 1) is a root of the equation X2 + 1 = 0.
Thus we have shown that there exists a field with the desired properties. Now we show that C0 is also the smallest such field.
Let F be any other field containing R such that X2 + 1 = 0 has a root in F. Take such a root and call it i1. Let C1 be the subfield {a + i1b: a, b ∈ R} of F. The representation a + i1b of the elements of C1 is unique, for
a + i1b = c + i1d ⇒
(a - c)2 = -(b - d)2 ⇒
a = c and b = d
Therefore the mapping f: C0 -> C1, f(a, b) = a + i1b is bijective. f preserves the field structure, so f is an isomorphism.
Thus any field containing R and a root of X2 + 1 = 0 also contains (a subfield isomorphic to) C0. C0 is therefore the smallest such field, so we may take C = C0.
Through the proof we have also learnt about the structure of C, which allows us to make some more definitions.
Definition:
Every z ∈ C has a unique representation a + ib where a, b ∈ R.
We call a, b the real and imaginary parts of z, and denote them by Re z, Im z respectively.
The complex conjugate of z is z* = a - ib.
The modulus of z is |z| = sqrt(a2 + b2).
With these elementary definitions we are in a position to build up a theory of complex numbers. In particular we can develop complex analysis, which probably provides the easiest proof of the fundamental theorem of algebra, i.e. that C is algebraically closed. That way C is shown to be the algebraic closure of R.