Felix' Math Place (Posts about fundamental theorem of algebra.)
https://math.fontein.de/tag/fundamental-theorem-of-algebra.atom
2019-11-17T10:38:27Z
Felix Fontein
Nikola
Fundamental Theorem of Algebra.
https://math.fontein.de/2009/05/04/fundamental-theorem-of-algebra/
2009-05-04T03:33:18+02:00
2009-05-04T03:33:18+02:00
Felix Fontein
<div>
<p>
As a warm-up, I want to give probably the most beautiful proof of the Fundamental Theorem of Algebra which I know, using the theory of one complex variable. In case you don't know the theorem:
</p>
<div class="theorem-environment theorem-theorem-environment">
<div class="theorem-header theorem-theorem-header">
<a name="fundamentalthm"></a>
Theorem (Fundamental Theorem of Algebra).
</div>
<div class="theorem-content theorem-theorem-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="65" height="18" src="https://math.fontein.de/formulae/FNfdtjBkcvuelsJ8a6wFTHdH1lzaXBlImmByog.svgz" alt="f \in \C[x]" title="f \in \C[x]"></span> be a polynomial, <span class="inline-formula"><img class="img-inline-formula img-formula" width="43" height="16" src="https://math.fontein.de/formulae/MIlLcY0A.raSgm4nDc_2iPnaSW_ZHoJi0dz1bg.svgz" alt="f \neq 0" title="f \neq 0"></span>. If <span class="inline-formula"><img class="img-inline-formula img-formula" width="75" height="16" src="https://math.fontein.de/formulae/XXZJJ93tO9nBs4l.hwrS6sbKcdkajfSuOYS2Ug.svgz" alt="n = \deg f" title="n = \deg f"></span>, there exist constants <span class="inline-formula"><img class="img-inline-formula img-formula" width="133" height="16" src="https://math.fontein.de/formulae/Ydo6a7kUuVhRsCCMXQgAu5lEnBRrhy32kXtcqg.svgz" alt="\alpha_1, \dots, \alpha_n, \beta \in \C" title="\alpha_1, \dots, \alpha_n, \beta \in \C"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="43" height="16" src="https://math.fontein.de/formulae/uJSbedd.H5dkq.SMdDd9W5EshGWxCPHPZS90Dw.svgz" alt="\beta \neq 0" title="\beta \neq 0"></span> such that
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="139" height="52" src="https://math.fontein.de/formulae/Ipcq136mybPVzXyivwAG0BOiwwuyLXaIl5q5UA.svgz" alt="f = \beta \prod_{i=1}^n
(x - \alpha_i)." title="f = \beta \prod_{i=1}^n
(x - \alpha_i).">
</div>
</div>
</div>
<p>
The main ingredient of the proof is the following statement, which is in fact eqiuvalent to the Fundamental Theorem:
</p>
<div class="theorem-environment theorem-lemma-environment">
<div class="theorem-header theorem-lemma-header">
<a name="fundlemma"></a>
Lemma.
</div>
<div class="theorem-content theorem-lemma-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="94" height="18" src="https://math.fontein.de/formulae/d23lzhdnqQ5lbODHthq3mtHduHDYeZguU1eVWA.svgz" alt="f \in \C[x] \setminus \C" title="f \in \C[x] \setminus \C"></span> be a non-constant polynomial. Then there exists an <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="13" src="https://math.fontein.de/formulae/qdxW81_tTHSfcFIOq4QhJ7OzDueW7Xa_sYG7pA.svgz" alt="\alpha \in \C" title="\alpha \in \C"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="68" height="18" src="https://math.fontein.de/formulae/gL7q0bnwus9JoiUbwy28T78BKlK.Y2In8Ws3Eg.svgz" alt="f(\alpha) = 0" title="f(\alpha) = 0"></span>.
</p>
</div>
</div>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<p>
Assume that on the contrary, <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/M.ji_f0zsVnTLyaKegBkRJLOeqrrt6brNnapOQ.svgz" alt="f" title="f"></span> is zero-free. In that case, <span class="inline-formula"><img class="img-inline-formula img-formula" width="91" height="24" src="https://math.fontein.de/formulae/jwfTQNKhe1mr.Rh6j7ET35oHMSwJBhr0SNDkGA.svgz" alt="g : z \mapsto \frac{1}{f(z)}" title="g : z \mapsto \frac{1}{f(z)}"></span> defines a entire function, i.e. a function defined on <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/sznF0_DZfMvBmQiFOcFDqhQZwWHPEvTQYYjJZA.svgz" alt="\C" title="\C"></span> which is holomorphic everywhere. We will show that <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="11" src="https://math.fontein.de/formulae/UxCAbnAvo.LhIVRUfLjfwIlPAfhwrVfYLg_Adw.svgz" alt="g" title="g"></span> is bounded, whence it follows by <a href="https://en.wikipedia.org/wiki/Liouville%27s_theorem_%28complex_analysis%29">Liouville's Theorem</a> that <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="11" src="https://math.fontein.de/formulae/UxCAbnAvo.LhIVRUfLjfwIlPAfhwrVfYLg_Adw.svgz" alt="g" title="g"></span> is constant. This implies that <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/M.ji_f0zsVnTLyaKegBkRJLOeqrrt6brNnapOQ.svgz" alt="f" title="f"></span> is constant, a contradiction.
</p>
<p>
First, write <span class="inline-formula"><img class="img-inline-formula img-formula" width="111" height="20" src="https://math.fontein.de/formulae/eVyidLoEydKuaeHkZB27H0clhkz9KArZkAHXtQ.svgz" alt="f = \sum_{i=0}^n a_i x^i" title="f = \sum_{i=0}^n a_i x^i"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="51" height="16" src="https://math.fontein.de/formulae/bTIbtUt5KwqEvh48.v_rnl0bSWi.nvUs.DtV8g.svgz" alt="a_n \neq 0" title="a_n \neq 0"></span>; then
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="247" height="44" src="https://math.fontein.de/formulae/UaIhbsHMysDhySggsJL_YmVLY8DqaFeGbFbaYg.svgz" alt="g(z) = z^{-n} \cdot \frac{1}{a_n
+ \sum_{i=0}^{n-1} a_i z^{i - n}}." title="g(z) = z^{-n} \cdot \frac{1}{a_n
+ \sum_{i=0}^{n-1} a_i z^{i - n}}.">
</div>
<p>
Clearly, for <span class="inline-formula"><img class="img-inline-formula img-formula" width="54" height="8" src="https://math.fontein.de/formulae/VJYOfKkIKkPBR8YA.ZTHGKosiqE1_sw8X3VQlA.svgz" alt="z \to \infty" title="z \to \infty"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="219" height="22" src="https://math.fontein.de/formulae/WJPdoQh3T_IUXcoNwHCMeoSGUiUvuyABal7w5A.svgz" alt="a_n + \sum_{i=0}^{n-1} a_i
z^{i - n} \to a_n \neq 0" title="a_n + \sum_{i=0}^{n-1} a_i
z^{i - n} \to a_n \neq 0"></span> uniformly, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="68" height="18" src="https://math.fontein.de/formulae/IjfAY7vyRICzALfyFgyvAWomZCTseDSz7RQVxw.svgz" alt="g(z) \to 0" title="g(z) \to 0"></span> uniformly for <span class="inline-formula"><img class="img-inline-formula img-formula" width="54" height="8" src="https://math.fontein.de/formulae/VJYOfKkIKkPBR8YA.ZTHGKosiqE1_sw8X3VQlA.svgz" alt="z \to \infty" title="z \to \infty"></span>. Therefore, <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="11" src="https://math.fontein.de/formulae/UxCAbnAvo.LhIVRUfLjfwIlPAfhwrVfYLg_Adw.svgz" alt="g" title="g"></span> is bounded on <span class="inline-formula"><img class="img-inline-formula img-formula" width="182" height="18" src="https://math.fontein.de/formulae/T0A8vJx0Zw9iGsa2doNYPkP7ZGp6MVgDopgw3Q.svgz" alt="B_1 := \{ z \in \C \mid \abs{z} > R \}" title="B_1 := \{ z \in \C \mid \abs{z} > R \}"></span> for some <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="13" src="https://math.fontein.de/formulae/3rzByTx4xORnQaqA2WwgTLGv7jy1L9T1w3rfrA.svgz" alt="R > 0" title="R > 0"></span>.
</p>
<p>
Now consider <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="11" src="https://math.fontein.de/formulae/UxCAbnAvo.LhIVRUfLjfwIlPAfhwrVfYLg_Adw.svgz" alt="g" title="g"></span> on <span class="inline-formula"><img class="img-inline-formula img-formula" width="182" height="18" src="https://math.fontein.de/formulae/x5i8aw.sVMO24mW0Uiv02QmPDCaqSRU8bMXW5g.svgz" alt="B_2 := \{ z \in \C \mid \abs{z} \le R \}" title="B_2 := \{ z \in \C \mid \abs{z} \le R \}"></span>. We have that <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="11" src="https://math.fontein.de/formulae/UxCAbnAvo.LhIVRUfLjfwIlPAfhwrVfYLg_Adw.svgz" alt="g" title="g"></span> is continuous on <span class="inline-formula"><img class="img-inline-formula img-formula" width="21" height="15" src="https://math.fontein.de/formulae/a4JodAAwTs0QGybCwZydrY0yZHo2jBdYpIjPOw.svgz" alt="B_2" title="B_2"></span>, and since <span class="inline-formula"><img class="img-inline-formula img-formula" width="21" height="15" src="https://math.fontein.de/formulae/a4JodAAwTs0QGybCwZydrY0yZHo2jBdYpIjPOw.svgz" alt="B_2" title="B_2"></span> is compact, we know that <span class="inline-formula"><img class="img-inline-formula img-formula" width="19" height="18" src="https://math.fontein.de/formulae/75dYHNy06kDvLK0UngNje_RnvByQ5GN.RK_HMQ.svgz" alt="|g|" title="|g|"></span> attains its maximum on <span class="inline-formula"><img class="img-inline-formula img-formula" width="21" height="15" src="https://math.fontein.de/formulae/a4JodAAwTs0QGybCwZydrY0yZHo2jBdYpIjPOw.svgz" alt="B_2" title="B_2"></span>, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="11" src="https://math.fontein.de/formulae/UxCAbnAvo.LhIVRUfLjfwIlPAfhwrVfYLg_Adw.svgz" alt="g" title="g"></span> is bounded on <span class="inline-formula"><img class="img-inline-formula img-formula" width="21" height="15" src="https://math.fontein.de/formulae/a4JodAAwTs0QGybCwZydrY0yZHo2jBdYpIjPOw.svgz" alt="B_2" title="B_2"></span>.
</p>
<p>
Therefore, <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="11" src="https://math.fontein.de/formulae/UxCAbnAvo.LhIVRUfLjfwIlPAfhwrVfYLg_Adw.svgz" alt="g" title="g"></span> is bounded on <span class="inline-formula"><img class="img-inline-formula img-formula" width="99" height="15" src="https://math.fontein.de/formulae/jLPh7RrBDPmuZEntQ1XeHU_fVRf2cUqDhHK5Eg.svgz" alt="B_1 \cup B_2 = \C" title="B_1 \cup B_2 = \C"></span>, and we can conclude.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
Now we are able to prove the Fundamental Theorem:
</p>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof (Fundamental Theorem).
</div>
<div class="theorem-content theorem-proof-content">
<p>
We proceed by induction on <span class="inline-formula"><img class="img-inline-formula img-formula" width="40" height="16" src="https://math.fontein.de/formulae/znBiRzj5K2MFAm57q44ue3ApWWEen9e9kU4k9A.svgz" alt="\deg f" title="\deg f"></span>, the degree of <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/M.ji_f0zsVnTLyaKegBkRJLOeqrrt6brNnapOQ.svgz" alt="f" title="f"></span>. If <span class="inline-formula"><img class="img-inline-formula img-formula" width="73" height="16" src="https://math.fontein.de/formulae/HuYaug8TTKxvN.NIfvbRSzLkUqVo1UVWW5p8Jg.svgz" alt="\deg f = 0" title="\deg f = 0"></span>, then <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="16" src="https://math.fontein.de/formulae/N7Pk9oWXc9tkT97wu2cjZSAUNTYEPc_yuqlK9A.svgz" alt="f \in \C" title="f \in \C"></span>, whence we can set <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="11" src="https://math.fontein.de/formulae/wLwTla43PAIjTiEW.RBdZLFel4yveL._6khsTA.svgz" alt="n := 0" title="n := 0"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="128" height="18" src="https://math.fontein.de/formulae/lgqjCm6kG4gkWhrCnd7rCg6eKqi1OorjafgQ_A.svgz" alt="\beta := f \in \C \setminus \{ 0 \}" title="\beta := f \in \C \setminus \{ 0 \}"></span>.
</p>
<p>
Now assume that the statement holds for polynomials of degree <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span>. Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/M.ji_f0zsVnTLyaKegBkRJLOeqrrt6brNnapOQ.svgz" alt="f" title="f"></span> be a polynomial of degree <span class="inline-formula"><img class="img-inline-formula img-formula" width="41" height="13" src="https://math.fontein.de/formulae/fgRY_0KZ1v9Dol4ARGUMspWZdQu629FDAEQ9MA.svgz" alt="n + 1" title="n + 1"></span>. By the <a href="https://math.fontein.de/2009/05/04/fundamental-theorem-of-algebra/#fundlemma">lemma</a>, there exists some <span class="inline-formula"><img class="img-inline-formula img-formula" width="73" height="16" src="https://math.fontein.de/formulae/_n2gjOWTGvAWkFKAgeNgMPHMZ6nlkxzaw24HYw.svgz" alt="\alpha_{n+1} \in \C" title="\alpha_{n+1} \in \C"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="96" height="18" src="https://math.fontein.de/formulae/ZUW5nlP3QtIS__Z3W1tbn8xTi52UzjC7u_4Lrg.svgz" alt="f(\alpha_{n+1}) = 0" title="f(\alpha_{n+1}) = 0"></span>. Now, using the <a href="https://en.wikipedia.org/wiki/Division_algorithm">Division Algorithm</a>, write <span class="inline-formula"><img class="img-inline-formula img-formula" width="170" height="18" src="https://math.fontein.de/formulae/L3k8bpb7ZAxgSssKlNwAYwh2hfuDlEoAch_4lw.svgz" alt="f = q \cdot (x - \alpha_{n+1}) + r" title="f = q \cdot (x - \alpha_{n+1}) + r"></span> with polynomials <span class="inline-formula"><img class="img-inline-formula img-formula" width="79" height="18" src="https://math.fontein.de/formulae/kwdoN3ywFMINqo_fLPRfpMwW01Oc5L9V9xcF3w.svgz" alt="q, r \in \C[x]" title="q, r \in \C[x]"></span> satisfying <span class="inline-formula"><img class="img-inline-formula img-formula" width="206" height="18" src="https://math.fontein.de/formulae/h6D7qAe0T2_y3DDvllmG0KK964OiJBXxVs6S0A.svgz" alt="\deg r < \deg (x - \alpha_{n+1}) = 1" title="\deg r < \deg (x - \alpha_{n+1}) = 1"></span>, i.e. <span class="inline-formula"><img class="img-inline-formula img-formula" width="43" height="13" src="https://math.fontein.de/formulae/j7LRYUKMIkIz50ugKik3J0EdukUs32MH7ODwVw.svgz" alt="r \in \C" title="r \in \C"></span>. Now
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="392" height="18" src="https://math.fontein.de/formulae/Mt5SU9JwtE2bUPcbCxC2ZoiNCKnrgLWezEKvwg.svgz" alt="0 = f(\alpha_{n_1}) =
q(\alpha_{n+1}) \cdot (\alpha_{n+1} - \alpha_{n+1}) + r = 0 + r," title="0 = f(\alpha_{n_1}) =
q(\alpha_{n+1}) \cdot (\alpha_{n+1} - \alpha_{n+1}) + r = 0 + r,">
</div>
<p>
whence we have <span class="inline-formula"><img class="img-inline-formula img-formula" width="41" height="11" src="https://math.fontein.de/formulae/7xNxpr3R49Q.4cLaK6bH0Avb9HKGrYQi4CuoPQ.svgz" alt="r = 0" title="r = 0"></span> and, therefore, <span class="inline-formula"><img class="img-inline-formula img-formula" width="140" height="18" src="https://math.fontein.de/formulae/lic3bBO9KXI8osYTC43bJBk0BGBYKbRWyt3JvA.svgz" alt="f = q \cdot (x - \alpha_{n+1})" title="f = q \cdot (x - \alpha_{n+1})"></span>. As <span class="inline-formula"><img class="img-inline-formula img-formula" width="328" height="18" src="https://math.fontein.de/formulae/H34qnMhgA7jYqKsFV.VizLHj4RCmc1WkY_iRqw.svgz" alt="\deg f = \deg q + \deg (x - \alpha_{n+1}) =
\deg q + 1" title="\deg f = \deg q + \deg (x - \alpha_{n+1}) =
\deg q + 1"></span>, we have <span class="inline-formula"><img class="img-inline-formula img-formula" width="73" height="16" src="https://math.fontein.de/formulae/ZD50PXbIFX6OvgdSIdfYxQh43bYkKby29MoOJQ.svgz" alt="\deg q = n" title="\deg q = n"></span>.
</p>
<p>
Therefore, by the induction hypothesis, there exist <span class="inline-formula"><img class="img-inline-formula img-formula" width="133" height="16" src="https://math.fontein.de/formulae/Ydo6a7kUuVhRsCCMXQgAu5lEnBRrhy32kXtcqg.svgz" alt="\alpha_1, \dots, \alpha_n, \beta \in \C" title="\alpha_1, \dots, \alpha_n, \beta \in \C"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="43" height="16" src="https://math.fontein.de/formulae/uJSbedd.H5dkq.SMdDd9W5EshGWxCPHPZS90Dw.svgz" alt="\beta \neq 0" title="\beta \neq 0"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="149" height="20" src="https://math.fontein.de/formulae/cgxw1gpcFcbhc1QIK9i11VArQh3zjDQzo10ysg.svgz" alt="q = \beta \prod_{i=1}^n (x - \alpha_i)" title="q = \beta \prod_{i=1}^n (x - \alpha_i)"></span>, whence
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="272" height="55" src="https://math.fontein.de/formulae/wkNQOiedNVpLj6iXLYtpM6I0Kv4iBFh8E6WuSA.svgz" alt="f = q \cdot (x - \alpha_{n+1})
= \beta \prod_{i=1}^{n+1} (x - \alpha_i)," title="f = q \cdot (x - \alpha_{n+1})
= \beta \prod_{i=1}^{n+1} (x - \alpha_i),">
</div>
<p>
i.e. the induction hypothesis holds for <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/M.ji_f0zsVnTLyaKegBkRJLOeqrrt6brNnapOQ.svgz" alt="f" title="f"></span>, too.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
</div>