Fundamental theorem of algebra
This shows that the field of complex numbers, unlike the field of real numbers, is algebraically closed. An easy consequence is that the product of all the roots equals and the sum of all the roots equals .
A weaker form of this theorem had been conjectured in the 17th century by Albert Girard. In his book L'invention en algèbre (published in 1629), he asserted that every polynomial equation of degree with real coefficients has solutions, but he did not state that they were necessarily complex numbers. A first attempt at proving the theorem was made by d'Alembert in 1746, but his proof was incomplete. Other attemps were made by Euler (1749), Lagrange (1772), and Laplace (1795). These last three attemps assumed implicitely Girard's assertion; to be more precise, the existence of solutions was assumed and all that remained to be proved was that their form was for some real numbers and .
A much more rigorous proof (which did not assume the existence of roots) was published by Gauss in 1799, but it had a topological gap. A rigorous proof was published by Argand in 1806; it was here that, for the first time, the fundamental theorem of algebra was stated for polynomials with complex coefficients, rather than just real coefficients. Gauss produced two other proofs in 1816 and another version of his original proof in 1849.
All proofs of the fundamental theorem necessarily involve some analysis, or more precisely, the concept of continuity of real or complex polynomials. The main difficulty in the proof is to show that every non-constant polynomial has at least one zero. We mention approaches via complex analysis, topology, and algebra:
- Find a closed disk of radius centered at the origin such that |p(z)| > |p(0)| whenever |z|≥r. The minimum of |p(z)| on D is therefore achieved at some point z0 in the interior of D. If |p(z0)| = m > 0, then 1/p(z) is a holomorphic function in the entire complex plane. Applying Liouville's theorem which states that a bounded entire function must be constant, we conclude that a polynomial without complex zeros must be constant. As an alternative to Liouville's theorem, we can take a Taylor series expansion of p(z) at z0: for some k > 0 and some non-zero constant ck, we have p(z) = p(z0) + ck(z − z0)k + ... It follows that for positive ε sufficiently small,
- For the topological proof by contradiction, assume p(z) has no zeros. Choose a large positive number R such that for |z| = R, the leading term zn of p(z) dominates all other terms combined. As z traverses the circle |z| = R once counter-clockwise, p(z), like zn, winds n times counter-clockwise around 0. At the other extreme, with |z| = 0, the "curve" p(z) is simply the single (nonzero) point p(0), which clearly has a winding number of 0. If the loop followed by z is continuously deformed between these extremes, the path of p(z) also deforms continuously. Since p(z) has no zeros, the path can never cross over 0 as it deforms, and hence its winding number with respect to 0 will never change. However, given that the winding number started as n and ended as 0, this is absurd. Therefore, p(z) has at least one zero.
- Replacing p(z) by its product with its complex conjugate, it suffices to check that the fundamental theorem is true for all polynomials with real coefficients. This can be proved by induction on the highest power of 2 dividing the degree of n. For n odd, a real polynomial of degree n has a real root by the intermediate value theorem. For n even, the number of two element subsets of an n element set is divisible by one less factor of 2 than n. We can therefore apply the induction hypothesis to the polynomials whose roots are given by symmetric functions in pairs of roots of p(z). If we know zi + zj and zizj are both complex numbers, then we can use the quadratic formula to show that zi and zj are in C.
- Remarks on Proving the Fundamental Theorem of Algebra
- The Fundamental Theorem of Algebra, Statement and Significance
- The Fundamental Theorem of Algebra for High School Students
- Benjamin Fine and Gerhard Rosenberger, The Fundamental Theorem of Algebra, 1997, Springer-Verlag, ISBN 0-387-94657-8
- Reinhold Remmert, “The Fundamental Theorem of Algebra”, in Numbers, 1991, Springer-Verlag, ISBN 0-387-97497-0
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