Theorem: (Fundamental Theorem of Algebra): Given any positive integer
$n \geq 1$ and any choice of complex numbers $a_0,a_1,…, a_n$, such
that $a_n \neq 0$, the polynomial equation$(1) \quad a_n z_n +···+ a_1 z + a_0 = 0$
has at least one solution $z \in \mathbb{C}$.
Proof extract: For our argument, we rely on the fact that the function
$|f(z)|=|a_n z^n+…+a_1 z+a_0|$ attains its minimum value. Let $z_0 \in \mathbb{C}$ be a point where the minimum is attained. We will show
that if $f(z_0) \neq 0$, then $z_0$ is not a minimum, thus proving by
contraposition that the minimum value of $|f(z)|$ is zero. Therefore,
$f(z_0) = 0$. If $f(z_0) \neq 0$, then we can define a new function $g : \mathbb{C} \to \mathbb{C}$ by setting$g(z)=\frac{f(z+z_0)}{f(z_0)}$, for all $z \in \mathbb{C}$.
Note that $g$ is a polynomial of degree $n$, and that the minimum of
$|f(z)|$ is attained at $z_0$ if and only if the minimum of $|g|$ is
attained at $z = 0$. Moreover, it is clear that $g(0) = 1$. More
explicitly, $g$ is given by a polynomial of the form$(2) \quad g(z)=b_n z^n+…+b_k z^k+1$
with $n \geq 1$ and $b_k \neq 0$ for some $1 \leq k \leq n$. Let
$b_k=|b_k| e^{i \theta}$, and consider $z$ of the form$(3) \quad z=r |b_k|^{-1/k} e^{i (\pi – \theta)/k}$,
with $r>0$. For $z$ of this form we have
$g(z)=1-r^k+r^{k+1} h(r)$
where $h$ is a polynomial.
The rest of the proof is not relevant for my questions.
i) How do we know that $g(z)$ has the form given in $(2)$? We know that $g(0)=1$, so it is clear that the constant must be $1$. It is also clear that the highest power is $n$ since we basically rescale $f(z)$ by a factor of $f(z_0)$ after translating it by $z_0$. But why must $g$ have this specific form? What would go wrong if we would simply let $k=1$? I think the main point is that when doing this transformation from $f$ to $g$ some powers with non-zero coefficients $a_j$ might drop out, but this cannot happen for $a_n$.
ii) The text mentions $b_k=|b_k|e^{i \theta}$, which I think is simply the representation of the complex number $b_k$ using polar coordinates. Similarly, $(3)$ specifies a complex number $z$ by its polar coordinates. Is this understanding correct?
This is the last bit I need to understand to finally get an understanding why the Fundamental Theorem of Algebra is true.
Thanks a lot for any help!
Best Answer
You are right about ii).
To answer question i), notice the requirement $b_k \ne 0$, which might or might not be true for $k=1$. So, that form which is written is not a specific form of $n$th degree polynomial satisfying $g(0)=1$, it's actually equivalent equivalent to the general form. You would start from the general form $g(z) = b_n z^n + ... + b_1 z + 1$ and then you would set $k = \min\{ i \mid 1 \le i \le n, b_i \ne 0\}$.