Group Theory and Linear Algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 27 September 2014

Lecture 32: Revision: The Fundamental Theorem of Algebra

(II) Week 1 §2: Show that the field of complex numbers is algebraically closed.

To show: If $f=a_0+a_{1}t+\cdots +a_{\ell }{t}^{\ell }\in ℂ\left[t\right]$ then there exists ${\beta }_1,{\beta }_2,\dots ,{\beta }_{\ell }\in ℂ$ such that $$a_0+a_{1}t+\cdots +a_{\ell }{t}^{\ell }=(t-{\beta }_1)\cdots (t-{\beta }_{\ell })\text{.}$$ Proof by induction on $\ell \text{:}$

If $f=a_0+a_{1}t+\cdots +a_{\ell }{t}^{\ell }\in ℂ\left[t\right]$ then there exist $\beta \in ℂ$ and $f_1=b_0+b_{1}t+\cdots +b_{\ell -1}{t}^{\ell -1}$ such that $$f=(t-\beta )f_1\text{.}$$

Another version of the Lemma is

If $f=a_0+a_{1}t+\cdots +a_{\ell }{t}^{\ell }\in ℂ\left[t\right]$ then there exist $\beta \in ℂ$ such that $$f\left(\beta \right)=0\text{.}$$

$ℝ$ is not algebraically closed: $t^2+1$ does not factor in $ℝ\left[t\right],$ even though $t^2+1=(t+i)(t-i)$ factors in $ℂ\left[t\right]\text{.}$

If $f=a_0+a_{1}t+\cdots +a_{\ell }{t}^{\ell }\in ℝ\left[t\right]$ then there exist $p_1,\dots ,p_r$ with $\text{deg}\left(p_i\right)$ equal to $1$ or $2$ such that $$f=p_1·p_2\cdots p_r\text{.}$$

This follows from the fundamental theorem of algebra and the fact that If $f\in ℝ\left[t\right]$ and $\beta \in ℂ$ such that $f\left(\beta \right)=0$ then $f\left(\bar{\beta }\right)=0\text{.}$ So $$\begin{array}{rcl}{\displaystyle f}& =& {\displaystyle (t-{\gamma }_1)\cdots (t-{\gamma }_r)(t-{\beta }_1)(t-\overline{{\beta }_1})\cdots (t-{\beta }_s)(t-\overline{{\beta }_s})}\\ & =& {\displaystyle (t-{\gamma }_1)\cdots (t-{\gamma }_r)(t^2-({\beta }_1+\overline{{\beta }_1}+{\beta }_1\overline{{\beta }_1}))\cdots (t^2-({\beta }_s+\overline{{\beta }_s})+{\beta }_s\overline{{\beta }_s})}\end{array}$$ with ${\gamma }_1,\dots ,{\gamma }_r\in ℝ$ and ${\beta }_1,\dots ,{\beta }_s\in ℂ$ with ${\beta }_1,\dots ,{\beta }_s\notin ℝ\text{.}$ Note that ${\beta }_j+\overline{{\beta }_j}\in ℝ$ and ${\beta }_j\overline{{\beta }_j}\in ℝ\text{.}$

This theorem is called the fundamental theorem of algebra. It was first proved by d'Alembert, after which Gauss studied the theorem intensively providing 14 proofs.

Further references: Wikipedia - Fundamental Theorem of Algebra, Math Overflow - Fundamental Theorem of Algebra, Article of Harm Derksen.

$ℂ$ is algebraically closed.

		Proof (d'Alembert-Gauss [Bou, Top. Ch VIII §1, no. 1, Theorem 1]).
	Bourbaki defines $ℂ$ as $ℝ\left[X\right](X^2+1)\text{.}$ To show: (a) If $a\in {ℝ}_{\ge 0}$ then there exists $\sqrt{a}\in ℝ\text{.}$ (b) If $p\left(t\right)\in ℝ\left[t\right]$ and $\text{deg}\hspace{0.17em}p$ is odd then there exists $\alpha \in ℝ$ such that $p\left(\alpha \right)=0\text{.}$ (b) Assume $p\left(t\right)=a_n{X}^n+a_{n-1}{X}^{n-1}+\cdots +a_0$ with $n$ odd and $a_n\ne 0\text{.}$ If $x\in ℝ$ and $x\ne 0$ then $$p\left(x\right)=a_n{x}^{n}g\left(x\right),$$ where $g\left(x\right)=1+\frac{a_{n-1}}{a_{n}x}+\cdots +\frac{a_0}{a_n{x}^n}\text{.}$ $$\underset{x\to \infty }{\text{lim}}g\left(x\right)=1\text{and}\underset{x\to -\infty }{\text{lim}}g\left(x\right)=1\text{.}$$ So there exists $a\in {ℝ}_{\ge 0}$ such that $$\text{sign}\left(a_n\right)=\text{sign}\left(f\left(a\right)\right)\text{and}\text{sign}(-a_n)=\text{sign}\left(f(-a)\right)\text{.}$$ Thus, by Bolzano's theorem, [Bou, Top IV §6, no. 1, Theorem 2], there exists $\alpha \in {[-a,a]}_{ℝ}$ such that $f\left(\alpha \right)=0\text{.}$ $\square$

		Proof 2 [Bou, Top. Ch VIII §2. Exercise 2].
	Let $f\left(t\right)\in ℂ\left[t\right]$ such that $f\left(t\right)\ne 0\text{.}$ To show: There exists $r\in {ℝ}_{\ge 0}$ such that if $z\in ℂ$ and $\left|z\right|\ge r$ then $\left|f\left(z\right)\right|>\left|f\left(0\right)\right|\text{.}$ Use [Exercise 1] and Weierstrass' theorem [Bou, Top. ChIV §6 no. 1, Theorem 1] to show $ℂ$ is algebraically closed. $\square$

[Bou Top. Ch. VIII §2, Exercise 1]
Let $a\in ℂ,$ $a\ne 0$ and $n\in {\mathbb{Z}}_{>0}\text{.}$

To show:

(a) If $r\in {ℝ}_{>0}$ such that $r^n\le \left|a\right|$ then there exists $z\in ℂ$ such that $\left|z\right|=r$ and $$|a+z^n|=\left|a\right|-r^n\text{.}$$ (b) If $f\left(z\right)\in ℂ\left[t\right]$ and $\text{deg}\left(f\right)>0$ and $z_0\in ℂ,$ with $f\left(z_0\right)\ne 0$ then there exists $z\in B_{\epsilon }\left(z_0\right)$ such that $$\left|f\left(z_0\right)\right|>\left|f\left(z\right)\right|\text{.}$$

Notes and References

These are a typed copy of Lecture 32 from a series of handwritten lecture notes for the class Group Theory and Linear Algebra given on October 19, 2011.

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