Page 81 - Linear Algebra Done Right

Complex Coefficients
deg r
j
in r and in cz p are equal).
choose c so that the coefficients of z
This contradicts our choice of s as the polynomial that produces the
smallest degree for expressions of the form q − sp, completing the 67
proof.
Complex Coefficients
So far we have been handling polynomials with complex coefficients
and polynomials with real coefficients simultaneously through our con-
vention that F denotes R or C. Now we will see some differences be-
tween these two cases. In this section we treat polynomials with com-
plex coefficients. In the next section we will use our results about poly-
nomials with complex coefficients to prove corresponding results for
polynomials with real coefficients.
Though this chapter contains no linear algebra, the results so far
have nonetheless been proved using algebra. The next result, though
called the fundamental theorem of algebra, requires analysis for its
proof. The short proof presented here uses tools from complex anal-
ysis. If you have not had a course in complex analysis, this proof will
almost certainly be meaningless to you. In that case, just accept the
fundamental theorem of algebra as something that we need to use but
whose proof requires more advanced tools that you may learn in later
courses.

4.7 Fundamental Theorem of Algebra: Every nonconstant polyno- This is an existence
mial with complex coefficients has a root. theorem. The quadratic
formula gives the roots
Proof: Let p be a nonconstant polynomial with complex coeffi- explicitly for
cients. Suppose that p has no roots. Then 1/p is an analytic function polynomials of
on C. Furthermore, p(z) →∞ as z →∞, which implies that 1/p → 0as degree 2. Similar but
z →∞. Thus 1/p is a bounded analytic function on C. By Liouville’s the- more complicated
orem, any such function must be constant. But if 1/p is constant, then formulas exist for
p is constant, contradicting our assumption that p is nonconstant. polynomials of degree
3 and 4. No such
The fundamental theorem of algebra leads to the following factor- formulas exist for
ization result for polynomials with complex coefficients. Note that polynomials of degree
in this factorization, the numbers λ 1 ,...,λ m are precisely the roots 5 and above.
of p, for these are the only values of z for which the right side of 4.9
equals 0.