D.3 The fundamental theorem of algebra
Remember our motivating example? We introduced the imaginary number i so that the equation x2 + 1 = 0 can have a solution. It turned out that complex numbers provide a solution for any polynomial equation.
Let’s introduce the set of polynomials with complex coefficients:
![{ } ∑ n k ℂ [x ] := ckx : ck ∈ ℂ, n ∈ ℕ0 . k=0](https://static.packt-cdn.com/products/9781837027873/graphics/media/file2214.png)
Analogously, ℝ[x],ℚ[x],ℤ[x] and ℕ[x] can be defined as well. The degree of a polynomial (or deg p for short) is the highest power of x. (For example, −3x8 + πx has a degree of 8.)
For a given polynomial p(x), solutions of the equation p(x) = 0 are called roots. Algebraically, it is desirable that for a given set of polynomials over a particular set of numbers, each polynomial has roots there as well. As we have seen, it is not true for ℝ[x], since x2 + 1 = 0 has no real solutions.
However, this changes for ℂ[x], as stated by the fundamental theorem of algebra.
Theorem 156. (The fundamental theorem of algebra)
Every...
