Templates

This is a basic article template, excerpted from the (at the time of this writing) current version of the article, Fundamental Theorem of Algebra.


Here is the source:

++ The Theorem

``A polynomial of positive degree over the field [[$\mathbb{C}$]] of complex numbers has a root in [[$\mathbb{C}$]].''[((bibcite clark))]

++ Overview

The polynomial [[$x^2 + 1$]] has no ``root'' in real numbers because [[$x^2 + 1 = 0$]] if and only if [[$x^2 = -1$]], and any real number squared equals [[$0$]] or a positive number.

Expanding one’s search to the complex numbers produces an answer, the square root of [[$-1,$]] also known as [[$\mathrm{i}:$]] [[$\mathrm{i}^2 = -1,$]] so [[$\mathrm{i}^2 + 1 = -1 + 1 = 0$]]. The Fundamental Theorem of Algebra says there is always a solution to a polynomial (that may have complex coefficients) if we're allowed to search through the complex numbers too.

A corollary (a relatively simple extension of the theorem) says that if the largest power in the polynomial is [[$n$]], then there are [[$n$]] complex solutions to a polynomial. However, some of the solutions may be repeats.

Clark[((bibcite clark))] cites Ankeny[((bibcite ankeny))] for his proof.

[[toc]]

++ Groundwork

**Notation and terms**

Following Clark's[((bibcite clark))] practice [[$fz$]] is written instead of [[$f(z)$]]. It's not particularly clear to me, however, why he then uses the parenthesized notation [[$\phi(z)$]] and [[$\psi(z)$]], although he does write [[$\phi \alpha$]].

[[math]]
fz = c_0 + c_1 z + \ldots + c_{n-1}z^{n-1} + c_n z^n \text{, where } n \geq 1
[[/math]]

[[math]]
\bar{f}z = \bar{c}_0 + \bar{c}_1 z + \ldots + \bar{c}_{n-1}z^{n-1} + \bar{c}_n z^n \text{, where } n \geq 1
[[/math]]

[[math]]
c = a + b\mathrm{i} \text{ if and only if } \bar{c}=a-b\mathrm{i}
[[/math]]

[[math]]
\phi = f\bar{f} = a_0 + a_1 z + \ldots + a_{2n} z^{2n}
[[/math]]

[[math]]
\phi = az^{2n} - \psi(z).
[[/math]]

[[math]]
-\psi(z) = a_0 + a_1 z + \ldots + a_m z^m, \text{ for } m<2n
[[/math]]

**Restate theorem**

If [[$f$]] is a polynomial of positive degree over the field [[$\mathbb{C}$]] of complex numbers then [[$f$]] has a root in [[$\mathbb{C}$]].

++ Assumption and Goal

**Assume**

[[$f$]] is a polynomial of positive degree over the field [[$\mathbb{C}$]] of complex numbers.

**Deduce**

[[$f$]] has a root in [[$\mathbb{C}$]].

[...]

because

[[math label888]]
\phi(z) = az^{2n} - \psi(z)
[[/math]]

[...]

(See also Eq. ([[eref label888]]).)

[...]

[[math label727]]
\frac{|a_0| + |a_1||z| + \cdots + |a_m||z|^m}{|a||z|^{2n}} \leq \frac{|a_0| + |a_1| + \cdots + |a_m|}{|a||z|^{2n-m}}
[[/math]]

because[[footnote]][...]Hence, comparatively speaking, the numerator is relatively larger than (or equal to) the denominator compared to before the change.[[/footnote]]

[...]

++ Notes

[[footnoteblock title=""]]

++ Sources

[[bibliography title=""]]
: ankeny : Ankeny, N.C. ``One more proof of the fundamental theorem of algebra.'' //Am. Math Monthly//, 54 (1947) 464, cited in Clark.
: clark : Clark, A. //Elements of Abstract Algebra//. New York: Dover, 1984 [orig. 1971].
: dennery : Dennery, P., and A. Krzywicki. //Mathematics for Physicists//. New York: Dover, 1996 [orig. 1967].
: irving : Irving, R.S. //Integers, Polynomials, and Rings//. New York: Springer, 2004.
: shankar : Shankar, R. //Basic Training in Mathematics: A Fitness Program for Science Students//. New York: Plenum, 1995.
[[/bibliography]]

 [[include end-material]]

[!-- remove initial space that was inserted so "include" code would show up--]

And here is how it's rendered:

The Theorem

“A polynomial of positive degree over the field $\mathbb{C}$ of complex numbers has a root in $\mathbb{C}$.”[2]

Overview

The polynomial $x^2 + 1$ has no “root” in real numbers because $x^2 + 1 = 0$ if and only if $x^2 = -1$, and any real number squared equals $0$ or a positive number.

Expanding one’s search to the complex numbers produces an answer, the square root of $-1,$ also known as $\mathrm{i}:$ $\mathrm{i}^2 = -1,$ so $\mathrm{i}^2 + 1 = -1 + 1 = 0$. The Fundamental Theorem of Algebra says there is always a solution to a polynomial (that may have complex coefficients) if we're allowed to search through the complex numbers too.

A corollary (a relatively simple extension of the theorem) says that if the largest power in the polynomial is $n$, then there are $n$ complex solutions to a polynomial. However, some of the solutions may be repeats.

Clark[2] cites Ankeny[1] for his proof.

Groundwork

Notation and terms

Following Clark's[2] practice $fz$ is written instead of $f(z)$. It's not particularly clear to me, however, why he then uses the parenthesized notation $\phi(z)$ and $\psi(z)$, although he does write $\phi \alpha$.

(1)
\begin{align} fz = c_0 + c_1 z + \ldots + c_{n-1}z^{n-1} + c_n z^n \text{, where } n \geq 1 \end{align}
(2)
\begin{align} \bar{f}z = \bar{c}_0 + \bar{c}_1 z + \ldots + \bar{c}_{n-1}z^{n-1} + \bar{c}_n z^n \text{, where } n \geq 1 \end{align}
(3)
\begin{align} c = a + b\mathrm{i} \text{ if and only if } \bar{c}=a-b\mathrm{i} \end{align}
(4)
\begin{align} \phi = f\bar{f} = a_0 + a_1 z + \ldots + a_{2n} z^{2n} \end{align}
(5)
\begin{align} \phi = az^{2n} - \psi(z). \end{align}
(6)
\begin{align} -\psi(z) = a_0 + a_1 z + \ldots + a_m z^m, \text{ for } m<2n \end{align}

Restate theorem

If $f$ is a polynomial of positive degree over the field $\mathbb{C}$ of complex numbers then $f$ has a root in $\mathbb{C}$.

Assumption and Goal

Assume

$f$ is a polynomial of positive degree over the field $\mathbb{C}$ of complex numbers.

Deduce

$f$ has a root in $\mathbb{C}$.

[…]

because

(7)
\begin{align} \phi(z) = az^{2n} - \psi(z) \end{align}

[…]

(See also Eq. (7).)

[…]

(8)
\begin{align} \frac{|a_0| + |a_1||z| + \cdots + |a_m||z|^m}{|a||z|^{2n}} \leq \frac{|a_0| + |a_1| + \cdots + |a_m|}{|a||z|^{2n-m}} \end{align}

because1

[…]

Notes

Sources

1. Ankeny, N.C. “One more proof of the fundamental theorem of algebra.” Am. Math Monthly, 54 (1947) 464, cited in Clark.
2. Clark, A. Elements of Abstract Algebra. New York: Dover, 1984 [orig. 1971].
3. Dennery, P., and A. Krzywicki. Mathematics for Physicists. New York: Dover, 1996 [orig. 1967].
4. Irving, R.S. Integers, Polynomials, and Rings. New York: Springer, 2004.
5. Shankar, R. Basic Training in Mathematics: A Fitness Program for Science Students. New York: Plenum, 1995.

Advisory

Please note that I am not a mathematician and so the presentation of proofs that I make may be deeply flawed. I'm using this writing process to figure out what I'm reading. Please consult more authoritative sources as well.

Feel free to contact me by leaving a comment or sending me a private message.

Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License