Complex integration · Definition
Fundamental theorem of algebra
Before you start you need to know
It says the complex plane is enough to solve any polynomial equation: no further extension of the numbers is needed.
Definition
Every non-constant polynomial \(p\) with complex coefficients, of degree \(n\), has at least one root in \(\mathbb{C}\); hence it splits into \(n\) linear factors \(z-z_k\), with leading coefficient \(a_n\).
The proof via Liouville is short. If \(p\) never vanished, \(1/p\) would be entire. Since \(|p(z)|\to\infty\) as \(|z|\to\infty\), \(1/p\) tends to \(0\) at infinity and is bounded on the whole plane, hence constant; but then \(p\) would be constant, against the hypothesis. So \(p\) has a root \(z_1\).
Once a root is found, division gives \(p(z)=(z-z_1)\,q(z)\) with \(q\) of degree \(n-1\). Repeating the argument exhausts the factors and gives \(p(z)=a_n(z-z_1)\cdots(z-z_n)\). Roots may repeat: the number of times \(z_k\) appears is its multiplicity, and the multiplicities add up to exactly the degree.
One says \(\mathbb{C}\) is algebraically closed, unlike \(\mathbb{R}\), where \(x^2+1\) has no root. If the coefficients are real, non-real roots appear in conjugate pairs \(z_k,\overline{z_k}\), so every real polynomial factors into real linear and irreducible quadratic factors. Vieta's formulas also relate roots and coefficients: the sum of the roots is \(-a_{n-1}/a_n\) and the product is \((-1)^n a_0/a_n\).
The theorem asserts that roots exist but not how to compute them; for degree \(\ge5\) there is no general formula with radicals. Typical mistakes: counting only distinct roots instead of counting multiplicities, or believing an odd-degree real polynomial has only real roots.
Think about it
Can a real polynomial of degree \(3\) have exactly one non-real root?
No: non-real roots come in conjugate pairs, so there are \(0\) or \(2\) non-real ones, and hence at least one real root.
Why does \(|p(z)|\to\infty\) as \(|z|\to\infty\) for a non-constant polynomial?
Because the highest-degree term dominates: \(|p(z)|\ge |a_n||z|^n/2\) for large \(|z|\).
Exercises 10
- How many complex roots, counted with multiplicity, does \(z^5-3z+1\) have?
- The roots of \(z^2+4\) are:
- What are the four roots of \(z^4-1\)?
- If \(p(z)=z^3+z+1\) has one real root, how many non-real roots does it have?
- The roots of \(z^3-6z^2+11z-6\) add up to:
- The product of the three roots of \(z^3+2z^2-z-10\) is:
- Which of these statements about polynomials and complex roots is true?
- The monic polynomial with roots \(1\) (double) and \(-2\) is:
- In the proof via Liouville, if \(p\) had no roots, which function would be entire and bounded?
- How many distinct roots does \(z^4-2z^3+z^2\) have?
References
- Elias M. Stein, Rami Shakarchi, Complex Analysis, 1st edition, Princeton University Press, 2003. Cap. 3, §4.3
- James Ward Brown, Ruel V. Churchill, Complex Variables and Applications, 9th edition, McGraw-Hill, 2014. §53
- Lars V. Ahlfors, Complex Analysis, 3rd edition, McGraw-Hill, 1979. Cap. 4, §3.2