$\mathrm{v}\mathrm{e}\mathrm{l}$. $*1 \mathrm{b}\mathrm{o}\mathrm{y}\mathrm{e}\mathrm{r}$ 1968] \mathrm{s}$ (TAJIMA, Toru)

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1 $\mathrm{v}\mathrm{e}\mathrm{l}$ $*1 \mathrm{b}\mathrm{o}\mathrm{y}\mathrm{e}\mathrm{r}$ 1968] \mathrm{s}$ (TAJIMA Toru) Graduate School of Arts and Sciences The University of Tokyo : $a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots$ $a_{1}x+a_{0}=0$ $\mathrm{n}$ $\mathrm{d}$ Alembert had spent much of his time and effort attempting to prove the theorem conjectured by Girard and known today as the fundamental theorem of algebra - that every polynominal equation $f(x)=0$ having complex coefficients and of degree $n>1$ has at least one com plex $]$ The statement which Gauss later referred to as the fundamentat theorern of algebra is $[$ root $\mathrm{d} \mathrm{a}1\mathrm{e}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{t} essentially the proposition known in France as theorem; but Gauss showed that all previously attempted demonstrations including some by Euler and Lagrange were inadequate $*1$ Struik : ( 1 ) (Demonstratio nova theoremis omnen functionem algebraicam rationalem integram unius variabilis in factores reales primi secundi grmdus resolvi posse) $*2$ 1 $*3$ 1 (1799) $*2$ $*3$ PP $[$$]$ [Gauss Werke] vol 3 p 73 Die lm Jahre 1799 erschienene Denkschrift Dernonstratio hatte einen doppelten Zweck n\"amlich erstens zu zeigen dass s\"amtliche bis dahin versuchte Beweise dieses wichtigsten Lehrsatzes der Theorie der algebraischen Gleichungen ungen\"ugend und illusonsch sind und zweitens einen neuen voilkommen strengen Beweis zu geben [Gauss 1799] [Gauss Werke] vol 3 pp 1-30

2 125 $\text{ }\ovalbox{\tt\small REJECT}_{\backslash }$ 1 $x^{n}+a_{1}x^{n-1}+\cdots+a_{n-1}x+a_{n}=0$ $\mathrm{f}\mathrm{f}\mathrm{l}1\re$ $f(x)=x^{n}+a_{1}x^{n-1}+\cdots$ +a lx+an $=0$ $\text{ }\mathrm{f}_{\backslash }$ $xt)^{\mathrm{p}}>\phi$ $\# 1$ ( ) $\text{ }\ovalbox{\tt\small REJECT}_{\backslash }$ $\text{ }\ovalbox{\tt\small $f(x)$ $f(x)$ $f(0)=a_{n}$ 0 $l\mathrm{h}a_{n}$ $\mathrm{c}$ REJECT}_{\backslash }$ $\mathrm{c}$ $f(x)-a_{n}$ $f(x)-a_{n}=x^{n}+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-1^{x}}$ $=x^{n}(1+ \frac{a_{1}}{x}+\frac{a_{2}}{x^{2}}+\cdots+\frac{a_{n-1}}{x^{n-1}})$ $ a + b \geq a+b $ $ a+b \geq a - b $ $ f(x)-a_{n} = x ^{n} 1+ \frac{a_{1}}{x}+\frac{a_{2}}{x^{2}}+\cdot$ $+ \frac{a_{n-1}}{x^{n-1}} $ $\geq x ^{n}\{1-\frac{a_{1}}{ x }-\frac{ a_{2} }{ x ^{2}}-\cdots-\frac{ a_{n-1} }{ x ^{n-1}}\}$ $\frac{ a_{1} }{ x }$ $\frac{ a_{2} }{ x ^{2}}$ $\text{ }*\mathrm{t}l\mathrm{f}$ $ x $ + 0 $\cdots\frac{ a_{n-1} }{ x ^{n-1}}$ $1- \frac{a_{1}}{ x }-\frac{ a_{2} }{ x ^{2}}-\cdots-\frac{ a_{n-1} }{ x ^{n-1}}\geq\frac{1}{2}$ $ f(x)-a_{n} \geq\frac{ x ^{n}}{2}> a_{n} $

3 $\text{ ^{}\prime}\mathrm{f}\mathrm{f}\mathrm{i}$ $\theta$ $=\mathrm{i}\mathrm{i}\mathrm{i}\rightarrow \mathrm{h}fl$ $\mathfrak{h}\mathrm{j}$ $\#$ $\text{ }\S\not\subset$ oe $\mathrm{a}$ $\ovalbox{\tt\small REJECT}-$ $\llcorner$ 126 $\ovalbox{\tt\small REJECT}_{\acute{\mathrm{f}}\backslash $ae\rightarrow\simeq$ $\text{ }\xi;\text{ }$ \mathrm{r}}\pm_{\mathrm{j}}$ $\text{ }\ovalbox{\tt\small REJECT}_{\backslash }$ $\text{ }\backslash \text{ }$ \mbox{\boldmath $\tau$}bffi E\Phi jf\not\in igg\re $\check{2}$ $\Phi 1\not\in$ $\text{ 1 }$ $\text{ }6_{\mathrm{i}}\mathrm{f}\mathrm{l}$ffl \sigma ) $\text{ }F\mathrm{f}\mathrm{l}$ 1 \Re -\check \doteqdot $f(x)$ 4 $\varpi\overline{\rightarrow}8fl$ $\Phi\tilde{\mathrm{x}}$ 4 $\mathrm{n}\beta\theta$ \Phi $\mathrm{c}$ $\text{ }\ovalbox{\tt\small REJECT}_{\backslash }$ $t\text{ }$ \Re a E V 1 $+4\neq$ ) $\xi$ E \mbox{\boldmath $\nu$}\ $\hslash^{\grave{\grave{1}}}$ 1 : \Gamma 4 (A $cta$ emdiforum) (Specimen novum analyseos pro scientia inflniti circa summas et quadraturas) \Xi \not\in - $*\backslash \backslash$g $\int dx$ : $(1+xx)$ (conflatus) [ ] $\overline{\tau\prime\backslash }$ $\check{j}\cdot*5$ $b$ $\cdots$ $c$ $x+b=l$ $x+c=m$ $x+d=n$ $l$ $m$ $n$ $\frac{\frac{\alpha}{\pi}+ex+;fxx+\frac{\delta}{\pi}x^{3}\pi\pi}{x^{3}+lx++^{\mu}xx+\frac{\lambda}{\pi}\pi\pi}=\frac{\frac{\alpha}{\pi}}{lmn}+\frac{\rho_{\frac{x}{\pi}}}{lmn}+\frac{\mapsto xx\tau \mathrm{r}}{lmn}+\frac{\frac{\delta x^{8}}{\pi}}{lmn}$ 1 $x$ 1 $(x+b=l)$ $\text{ }2$ $\frac{x}{lmn}$ $\frac{x^{2}}{lmn}=\frac{1}{n}-\frac{b}{lmn}$ $*4$ [Gauss 1799] pp $2n$ $\theta=4\pi/n$ $\hslash_{1}$ $3\theta$ $(8n-3)\theta$ $(8n-1)\theta$ $4n$ $P_{0}$ $P_{1}$ $f\backslash$ $P_{4n-1}$ F ffi ae\re { $\ldots$ $\ldots$ $\psi\backslash$ $\mathrm{v}\circ $*5\{\mathrm{L}\mathrm{M}\mathrm{G}$] 15$ p 351; 3 p 2o8

4 $\mathrm{f}$ $\ovalbox{\tt\small REJECT}$ REJECT}$ $\mathrm{a}$ $\cdot$ $x$ $\frac{x^{2}}{lmn}=\frac{1}{n}-\frac{b+c}{mn}+\frac{b^{2}}{lmn}$ $\frac{x^{3}}{lmn}=1-\frac{b+c+d}{n}+\frac{b^{2}+\mathrm{c}^{2}+bc}{mn}-\frac{b^{3}}{lmn}$ \mbox{\boldmath $\tau$}\\not\in 4 1 \not\in Fa $\ovalbox{\tt\small 1 $\not\cong\re$ ig\phi g $\hslash^{\mathrm{i}}\text{ }$ $*6$ $\int\frac{dx}{x^{4}+a^{4}}$ $\frac{1}{x^{4}+a^{4}}=(x+a\sqrt{\sqrt{-1}})(x-a\sqrt{\sqrt{-1}})(x+a\sqrt{-\sqrt{-1}})(x-a_{1^{\frac{\ovalbox{\tt\small REJECT}_{-}}{\sqrt{-1}})}}1$ $\llcorner$ $\hslash\not\in \text{ }$ dplcl 4 ( ) \mathrm{p}\mathrm{p}\mathrm{f}\hat{\mathrm{f}\mathrm{l}}}^{\prime\supset\lambda}$ 4$\beta_{\backslash $rx\text{ }$ (2 ) $*7$ $(x+a\sqrt{\sqrt{-1}})(x+a\sqrt{-\sqrt{-1}})=x^{2}+\sqrt{2}ax+a^{2}$ $1$ $\mathrm{a}\check{\vee J}$ # H AD b $\sim 18$ { 17 $ \backslash$ [ 1 { / $\grave{\mathrm{f}}\mathrm{g}\ovalbox{\tt\small REJECT}$ ffl L]\Xi J $n$ 20 $\backslash $\Rightarrow \mathrm{a}\mathrm{w}\mathrm{f}\mathrm{l}\mathrm{f}\mathrm{f}$ altiorum gradum) (1743) /fi\mathrm{i}$ $\mathrm{b}^{\mathrm{a}}$ $\text{ }r\mathrm{a}\mathrm{e}$ (De integratione aequationum differentialium $Ay+B \frac{dy}{dx}+c\frac{d^{2}y}{dx^{2}}\cdots+n\frac{d^{n}y}{dx^{n}}=0$ (1) $\grave{\grave{1}}$ $\text{ }$ $\text{ }$ $\text{ }Pz$ ( $p$ \not\in $y=e^{px}$ ) $A+Bp+Cp^{2}\cdots+Np^{n}=0$ (2) $*6$ ibid $\mathrm{p}\mathrm{p}$ $*7[\mathrm{L}\mathrm{M}\mathrm{G}]$ vol $361\sim \mathrm{s}53\mathrm{f}$ : $\mathrm{p}\mathrm{p}$ $209$ 21if $\mathrm{i}$ 5 p359; 1999] p 219

5 ] $1_{l}$ 128 $y=e^{\mathrm{p}x}$ $Ae^{px}+B \frac{de^{px}}{dx}+c\frac{d^{2}e^{px}}{dx^{2}}\cdots+n\frac{d^{n}e^{px}}{dx^{n}}=0$ $\rho_{\backslash -} \ovalbox{\tt\small \llcorner rb $n$ REJECT}$ $p=e^{px}$ $1_{\mathit{1}}$ $A+Bp+Cp^{2}\cdots+Np^{n}=0$ ff g *j^e -R- \not\in )fflj $\iota\backslash$ ( ) ] (1) $\ovalbox{\tt\small REJECT}$ (Introductio in $\text{ }$ analysin infinitomm) Effl $\mathrm{t}\backslash$ (De transformatione functionum) $:i^{r}$ $\theta^{\mathrm{i}\beta}fl$ 7p \hslash \not\in \Re : $z$ $z$ $n$ $n$ 2 3 $*8$ $\pi\overline{\overline{-}}$ 8fl $\text{ }$ $v\backslash$ 4 31 ; $Q$ 4 fflfm (factores simplices imaginaris) $Q$ (factores duphces reales) 9 $Q$ $z^{4}+az^{3}+bz^{2}+cz+d$ 2 $z^{2}-2(p+q\sqrt{-1})z+r+s\sqrt{-1}$ $\mathrm{a}\backslash$ $z^{2}-2(p-q\sqrt{-1})z+r-s\sqrt{-1}$ 2 (factores duplices imaginaries) 1 [ ] $z^{4}+az^{3}+bz^{2}+cz+d$ [ [ 2 $*9$ (I) (H) (III) (IV) 2001] p 18 $*8$ JEuler Opera) (1) vol 7 p 34 ; [ $*9$ ibid $\mathrm{p}\mathrm{p}$ : pp 19-20

6 \mathrm{j}l^{f}\mathrm{c}^{-}\ovalbox{\tt\small REJECT}$ $1+$ $4\mathrm{f} \simeq\xi\backslash \eta 2^{\cdot}\text{ }\sigma\supset \text{ }$ }$ 128 $\}_{\llcorner}^{r_{4}}\text{ }$ $z$ $(\mathrm{i})(\mathrm{i}\mathrm{i}\mathrm{i})$ $\text{ _{}\mathrm{r}\backslash }\rho$ $(t=p^{2}-q^{2}-r u=2pq-s)$ \check 2 $\theta $\text{ }\text{ }$ \mathrm{i}\mathfrak{x}$ $\hslash \mathrm{l}_{\mathrm{d}}^{\mathrm{a}}$ $\text{ }$ ffiae $+qq-p\sqrt{2t+2\mapsto tt+uu}-q\sqrt{-2t+2\sqrt{tt+uu}}\sqrt{tt+uu}$ $(\mathrm{i}\mathrm{i})(\mathrm{i}\mathrm{v})$ 2 $Q$ $\varphi_{\wedge}\backslash \mathrm{f}\mathrm{f}\backslash \mathrm{j}$ { 2 \S \supset 2 2 $\not\cong $:\exists 8fl$ $\urcorner \mathrm{p}\phi \mathrm{r}$ 2 ge+ $*10$ \text{ }$ $\mathrm{i}\xi 4 \text{ }$ 2 \not\equiv -\check -\rightarrow \beta E \mathrm{u}}$ $\mathrm{a}\mathrm{e}\text{ }$ $\mathrm{j}\backslash \mathrm{e}$ $\mathrm{f}\mathrm{f}\mathrm{i}^{\prime 32 $l\mathscr{l}\text{ }$ $\iota\backslash$ $\mathrm{f}\mathrm{f}\mathrm{i}^{\underline{\simeq}}$ 32 Bfl $\mathrm{f}\mathrm{f}\mathrm{i} 1\not\in$ $2n$ $n$ 2 $\text{ }$ $4^{\backslash $\text{ }\backslash \text{ }$ $8fl\not\in\exists$ lfffl $z$ $\mathrm{f}\mathrm{f}\mathrm{i} $E 2 $\overline{\overline{\simeq}}\mathrm{a}$bfl $\text{ }1\mathrm{E}$ $\iota\backslash$ $a$ $bz^{n}$ $a+bz^{n}+cz^{2n}$ $a+bz^{n}\dotplus cz^{2n}+dz^{3n}$ $\cdots$ $/i\theta\hslash\not\in$ \not\cong 2 $\overline{\mathit{2}}\cdot*11$ $\text{ $\overline{\equiv}\mathrm{e}^{\backslash }\mathrm{w}\mathrm{f} \backslash$ 5 }$ ( \Re # ) X $tf]$ $k^{\mathrm{m}}${ $\#_{arrow}^{\vee}\lambda\backslash$: \Rightarrow $r+_{\backslash }\cdot $2n$ $=\mathrm{j}\beta \mathrm{e}\s fl$ \text{ ^{}\prime}$ $\mathrm{f}\mathrm{f}\mathrm{l}\mathrm{i}\zeta$ $\mathrm{f}\mathrm{f}_{-\backslash }^{\mathrm{g}}$ $f\backslash$ $\mathrm{f}\mathscr{f}+_{\backslash }\text{ }\backslash$ $n$ 2ffiaej {fbf- $\mathrm{f}\mathrm{f}\backslash ffl \Xi nar 1740 {\lambda --#xm \equiv nj]ihfl $\text{ }$ $\Leftrightarrow--\vec{\overline{\emptyset}\mathrm{f}\mathrm{f}\mathrm{f}\mathrm{l}}$ 4 $\pi\overline{\simeq}\text{ }$ $1*r\mathrm{a}\ovalbox{\tt\small REJECT} \mathrm{e}$ $\iota\backslash$ 4 (2) $\text{ }$ $10$ ibid p36 : p ibid p 37 : PP $\mathrm{g}\text{ }$ $\mathrm{t}\backslash$ (De investigatione factorum trinominalium) {

7 130 -\rho fi \iota $\text{ }$ $n$ $\ovalbox{\tt\small REJECT}- \mathrm{f}\mu_{\mathrm{l}\mathrm{b}}\wedge$ $\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$ $(\cos\phi$ $\sqrt{-1}\sin\phi)^{n}=\cos n\phi\pm\sqrt{-1}\sin n\phi$ \sigma )\not\in ffl $\llcorner$ $a^{n}\pm z^{n}$ $\alpha+\beta z^{n}+\gamma z^{2n}$ $*12$ $\nu\backslash$ $k^{h}$ 154 : $\overline{\mathrm{w}}^{\mathrm{a}}\acute { $\mathrm{k}^{\mathrm{m}}$ \mathrm{w}\text{ }$ $\alpha+\beta z^{n}+\gamma^{2n}+\delta^{3n}$ $n+\theta z^{n}$ g $\iota+\chi z^{n}+\lambda z^{2n}$ $\iota\backslash$ $\mathrm{v}$ (multipiicator) \acute x \simeq Rlf E} $\Pi\overline{\mathrm{p}}\text{ }$ (in factores resolvi) $*13$ (1) (2) 2 4 (I) $\alpha+\beta z^{n}+\gamma z^{2n}+\delta z^{3n}$ (II) $\alpha+\beta z^{n}+\gamma z^{2n}+\delta z^{3n}+\epsilon z^{4n}$ (III) $\alpha+\beta z^{n}+\gamma z^{2n}+\delta z^{3n}+\epsilon z^{4n}+\zeta z^{5n}$ (1) (2) $*14 $ Gilain V *15 2 (Recheoehes sur les mcines imaginaires des \ equations)\rfloor (1749) 1751 $\overline{512153\mathrm{f}\mathrm{f}\mathrm{l}^{\text{ }5 \theta \S 8\text{ }\Resupset*\iota \text{ ^{}g)\mathrm{t}\mathrm{f}a^{10}-2a^{5}z^{5}\cos g+z^{10}}}}=(a^{2g}-2az\cos+z^{2})5(a^{2}-2az\cos \mathfrak{f}+z^{2})\langle a^{2}-2az\cos \mathfrak{f}+$ $z^{2})(a^{2}-2az\cos-\underline{4\pi}_{\overline{5}}-b+z^{2})(a^{2} - 2az\cos\underline{4\pi}_{5}\pm \mathrm{a}+z^{2})$ PP ibid P 164; [Euler Opera] (1) vo18 PP : p ibid PP ; $\mathrm{p}\mathrm{p}$i : Quare si ullum dubium mansisset circa huiusmodi resolutionem omniurn functionum integrarum hoc nunc fere penitus tolletur 15 $\mathrm{p}$ [Gilain 1991] 110

8 $\delta$ i20 et }$ toutes ne $\fbox_{\mathrm{p}}\pi$ $\ovalbox{\tt\small REJECT}- \text{ }$ $\mathrm{a}$ op\ erations $\overline{\mathrm{i}\overline{-}}\mathrm{k}\text{ }$ $\text{ }$ { $\mathfrak{b}\mathrm{e}$ MF/b $\ovalbox{\tt\small REJECT} $\text{ }\mathrm{a}_{\rangle\backslash \mathrm{r}$ 19 { * \neq \hslash \acute X $\mathrm{t}\backslash$ $\text{ }\mathrm{a}\mathrm{e}$ $\mathrm{v}\backslash$ R$ \mbox{\boldmath $\delta$}> 7\neq - $\Psi^{\mathrm{J}} \iota \mathrm{z}j4$ \ni \beta max \mbox{\boldmath $\tau$} $4\backslash$ [Euler Fuss] [Euler Opera Post] $\text{ }$ }Jffl $\eta \mathrm{p}$ j/\star \Phi \Phi ( ) $\mathfrak{s}\mathrm{f}\mathrm{f}\mathrm{i}$ 2 { $\text{ $\ovalbox{\tt\small REJECT}_{\overline{ \nu}}^{\overline{-}}\#\mathrm{c}$ }$ $M+N\sqrt{-1}$ ( $M$ $N$ ) $\text{ }\grave{/}\#$ $\overline{\overline{\vec{\overline{\mathfrak{n}}}}}^{\mathrm{t}}\llcorner$ Bfl \iota $M+N\sqrt{-1}$ $k^{\backslash }A$ $\equiv-\mathrm{r}\mathrm{p}\mathrm{b}fl$ E $\mathrm{f}\mathrm{l}_{\backslash }$ g ( ) $\ovalbox{\tt\small REJECT}-\ni\ovalbox{\tt\small REJECT}$ 2 $1_{\mathit{1}}\backslash$ \mathrm{f}\mathrm{f}\mathrm{i}\text{ }$ $T+_{\tau}\text{ }$ $\text{ }\backslash$ $\text{ _{}\mathrm{i}}\mathrm{x}\mathrm{r}$ 1 \Gamma \llcorner - --\Re $\text{ }\#$ $7\mathrm{f}\mathrm{f}\mathrm{i}\yen \text{ }\mathrm{h}$ $\backslash $y_{\backslash 2 (}3\text{ }$ ) ( ) $*17$ $\text{ }\mathrm{a}\mathrm{e}$ $\text{ }\backslash \cdot\phi\backslash$ $*16$ $\mathrm{a}\backslash$ lfftb \Phi }$ $\#)_{r}\mathrm{f}\mathrm{f}\mathrm{i}_{\backslash \backslash }$ $\ni \mathrm{a}\mathrm{r}\mathrm{r}s\text{ }$ $\mathrm{z}\hat{6}8fl$ $\text{ $=^{\mathrm{a}}\mathfrak{g} \mathrm{w}\text{ }$ $\Phi $\ovalbox{\tt\small REJECT}\vec{\frac{}{\mathrm{R}}}$ Bfl \mathrm{f}\mathrm{f}\mathrm{l}$ \Rightarrow $\ovalbox{\tt\small REJECT} ( fflffl ) \mathrm{f}\mathrm{f}\mathrm{i}4${ i^\epsilon \Phi --Q--jEBfl $\acute{l}\not\in fj$ $\text{ $\text{ }\backslash \mapsto\backslash $\Re_{\mathrm{R}}^{\mathrm{A}}\mathrm{g}$ +$ 5 }$ $\urcorner \mathrm{p}\hslash\geq 4$ 4\ [Euler 1751] $\llcorner$ 2 \check 7g $\Gamma+_{\backslash }\text{ }$ 4 $\text{ }\mathrm{f}\mathrm{f}\mathrm{i}$ 4 \not\in \Gamma -pj ffi g 2&\E $\mathrm{f}\mathrm{f}\mathrm{a}^{\mathrm{f}}\hslash\backslash$ $x^{3}$ 4 $x^{4}+bx^{2}+cx+d=0$ (3) $\alpha$ $\beta\gamma$ $x^{3}$ (3) 0 $\alpha+\beta+\gamma+\delta=0$ (4) $\text{ }$ (3) 1 (le premier membre) ( $\overline{*16}$ ${ [Euler1751]p121 les racinesi $\mathrm{m}$aginaires $\sim 17$ $(x^{2}-ux+\mu)(x^{2}+ux+\lambda)$ (5) $($ $)$ ne contiennent point $\mathrm{d} \mathrm{a}\mathrm{u}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{s}$ des raclnes outre les quatre op\ erations vulgaires $\mathrm{s} \mathrm{y}$ saurait soutenir que des op\ erations transcendantes m\^elassent ibid $\mathrm{p}$ : $($ $)$ $1 \mathrm{o}\mathrm{n}$ que l extraction

9 $\Pi \mathrm{p}$ 132 $u$ (3) 2 { $\text{ }$ $u$ 6 $u$ $F_{6}(x)=0$ $\backslash [ ${}_{4}C_{2}=6$ \mathrm{f}\mathrm{f}\mathrm{l}f_{-} $ # \pm p\rightarrow -paw It $(u)=0$ (6) $\overline{--\mathrm{j}-}\xi Hfl$ Xg (6) 1 $\cup$ }\backslash \text{ }\mathrm{g}_{1}^{\theta}\mathrm{g}$ $u$ \Re } \check c $\mu_{\backslash $u_{1}=\alpha+\beta$ $u_{2}=\alpha+\gamma$ $u_{3}=\alpha+\delta$ $u_{4}=\gamma+\delta$ $u_{5}=\beta+\delta$ $u_{6}=\beta+\gamma$ $u_{4}=-u_{1}$ $u_{5}=-u_{2}$ $u_{6}=-u_{1}$ (6) $F_{6}(u)=(u^{2}-u_{1}^{2})(u^{2}-u_{2}^{2})(u^{2}-u_{3}^{2})=0$ $\hslash_{\backslash }\mathrm{a}\mathrm{e}$ $u_{1}^{2}u_{2}^{2}u_{3}^{2}$ $\mathrm{g}\text{ }$ \equiv p-je\beta fl $u_{1}^{2}u_{2}^{2}u_{3}^{2}$ $\text{ }\ovalbox{\tt\small $\alpha\beta\gamma\delta$ :Fg (3) \Phi [ REJECT}_{\yen}^{\mathrm{D}}$ (3) $\pi^{\wedge}\backslash \mathrm{x} \backslash$ $\mathfrak{o}$ (4) \breve \check 2\not\in { $\text{ }t\mathrm{h}$ $u_{1}u_{2}u_{3}$ $u_{1}u_{2}u_{3}$ \sigma \supset \Supset pn\delta B fflj pfl\hslash \not\in \Re \Delta $\mathrm{p}_{\mathrm{h}}^{\backslash 4\ }$fi $\ovalbox{\tt\small $=x\text{ }\mathrm{p}\re$ $-\vee$ a\supset 5 4 REJECT}_{\lrcorner_{\mathrm{i}}}^{\{}\mathrm{f}\mathrm{P}\mathrm{J}$ $\sigma$) $\mathrm{a}\veearrow\wedge$bfl p $\ovalbox{\tt\small REJECT}^{r}\backslash $l\mathrm{t}^{\backslash }$ $\overline{\equiv}\pi^{\mathrm{p}}\mathrm{f}\mathrm{l}$ \mathfrak{h}$ $\backslash \backslash $\varphi_{\mathrm{e}} \ovalbox{\tt\small REJECT}- \mathrm{c}^{\theta}$ \mathrm{f}\mathrm{f}\mathrm{i}$ 19 $\mathrm{g}_{\check{\mathrm{x}}_{-}}$ $\ovalbox{\tt\small REJECT} \mathrm{f}\mathrm{f}\mathrm{i}\text{ }$ \mathrm{r}\mathfrak{n}\mathrm{r}\mathfrak{b}$ $\Supset\Delta PR5 2 $\mathrm{i}\underline{-}\mathrm{f}^{\mathrm{b}fl}$ $\square \ovalbox{\tt\small \acute x REJECT}-$ J $\vec{\frac{}{\rightarrow}}ae$ gfl $\mathfrak{p}\equiv-\mathrm{j}$]$\mathrm{i}8fl$ 1 \Phi $f $ 18

10 133 [1745] : $f3;$ $*18$ $\mathrm{r}^{1}1\mathrm{f}\mathrm{f}\mathrm{i}$ a 3: : $(a+b\sqrt{-1})^{m+n\sqrt{-1}}=x+y^{\sqrt{-1}}$ $(m+n \sqrt{-1})\frac{da+db\sqrt{-1}}{a+b\sqrt{-1}}=\frac{dx+dy\sqrt{-1}}{x+y\sqrt{-1}}$ ifi $a-b\sqrt{-1}$ $x-y\sqrt{-1}$ $(m+n \sqrt{-1})\frac{adb+bdb\sqrt{-1}+ada+bdb}{aa+bb}$ $= \frac{xdx+ydy+(xdy-ydx)\sqrt{-1}}{xx+yy}$ $\log\ovalbox{\tt\small REJECT} xx-yy=m\log\sqrt{aa+bb}-n\int\frac{adb-bda}{aa+bb}$ $\int\frac{xdy-ydx}{xx+yy}$ $=mf \frac{adb-bda}{aa+bb}+n\log\sqrt{aa+bb}$ $y$ $x$ $\mathrm{g}$ $y$ $x$ \sigma ) $r$ $r=c^{m\log\sqrt{aa+bb}-nf\frac{d}{\alpha}}\underline{a}b-bdaa\ovalbox{\tt\small REJECT}+=(\sqrt{aa+bb})^{m}\mathrm{x}c^{-nfarrow\div_{+}^{-b}}$ $m \int\frac{adb-bda}{aa+bb}+n\log\sqrt{aa+bb}$ $*19$ \leq 4 { (valeur analytique) $xy$ $*20$ 7 $f\tilde{}$ $ffi\backslash$g a f= $4\neq \mathrm{i}_{\iota}^{\mathrm{b}}\mathrm{s}_{\backslash }$ (fonction quelconque) (grandeur) 7 $x+y\sqrt{-1}$ $p_{1}^{\mathrm{p}}$ Bfl $\text{ }\acute{\mathrm{t}}^{\mathrm{a}}\yen$ $p+q\sqrt{-1}$ $*18F_{7\sqrt[\backslash ]{}\wedge^{\backslash }}^{arrow}$ $J\vee \text{ }$ \acute \llcorner { Chriitian Gilain JGiiain $1991\underline{\rceil}\mathrm{p}\mathrm{p}$ $ $ G \mbox{\boldmath $\nu$} $*19$ $f \frac{adb-bda}{aa+bb}$ $*20$ [Gilain 1991] $\mathrm{p}\mathrm{p}$ $ $ $d( \frac{b}{a})/(1+\frac{bb}{aa})$ \iota \ \supset {

11 $\lfloor(\mathrm{b}\mathrm{o}\mathrm{y}\mathrm{e}\mathrm{r}$ 1968] S\"amtliche $\mathrm{s}\mathrm{p}\pi//$ $\hat{\lambda \mathrm{e}}\text{ }$ $1_{1}\backslash$ $\prime A^{\backslash }\backslash \#$ Leibzig $\iota\backslash$ $\mathcal{t}\mathrm{g}$ 134 $p+q\sqrt{-1}$ \tilde a $p-q\sqrt{-1}$ $\mathrm{f}\mathrm{b}\emptyset\grave{\grave{:}}l\mathrm{f}\backslash \text{ }$ $\mathrm{h}$ $\Pi\overline{\mathrm{p}}\#\not\equiv$ 1 ( 9) ] $\mathscr{f}\text{ }$ { \mbox{\boldmath $\tau$}\mp l /p\\mbox{\boldmath $\zeta$}l\rightarrow -\beta Am ( 1 $\ovalbox{\tt\small REJECT}- 0) A\hat{\mathfrak{W}}$ 17 $\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}_{\vee}$ 18 $6fi$ 18 g\phi $\mathrm{f}_{\mathrm{p}}8\mathrm{f}\mathrm{f}\mathrm{l}$ $\mathbb{h} \text{ }$ ff $\ovalbox{\tt\small REJECT} \text{ }t_{\mathrm{l}} \S \mathrm{j}$ $\not\in $*\mathrm{e}^{\mathrm{r}}\mathrm{r}^{1}\mathrm{j}$ 19 8\text{ }$ { \not\cong $\#^{\mathit{1}}\mathrm{e}^{\cdot}\text{ }$ 18 $\not\in $\text{ ^{}\backslash }$ffi g \mathrm{f}\mathrm{f}\mathrm{i}$ 17 $\ovalbox{\tt\small REJECT} $\check{\mathrm{x}}_{-}$ $\vee\zeta^{\backslash }\text{ }$ $\mathrm{g}\ovalbox{\tt\small REJECT}$ $\rho_{+}*\mathrm{f}\mathrm{l}$ $\mathrm{b}^{\cdot}$ffl 19 ae 18 \mathrm{f}\pi$ $[BM]$ Bibliotheca mathematica: Zeitschreft f\"ur Geschichte der Mathernatik Dritte $\mathrm{f}\mathrm{o}1\mathrm{g}\mathrm{e}_{7}$ [LMG] Leibnizens Mathematische Schnften hrsg von C I Gerhardt Halle ; (Hildesheim-New York 1971)(rep) Schreften und Briefe hrsg von Preussishe Akademie der Wissenschaften zu $\mathrm{g}\mathrm{w}$ [LSB] Leibniz Berlin und von der Deutschen Akademie der Wissenschaften der DDR Darmstadt Berlin [MARS] Mdmoires de royale des sciences de Parw $l acad\acute{e}mie$ $[StL]$ Studia Leibnitiana : Zeitschnft f\"ur Geschichte der Philosophie und der Wissenschaften Wiesbaden 1969 [Aiton 1985] Aiton $\mathrm{e}\mathrm{j}$ Leibniz :a biography (Bristol : Hilger 1985) [Bachm acova 1960] Bachmacova Isabella Le th\ eor\ eme de l algebre et la construction des corps algebriques Archives internationales histoire des sciences 13(1960) pp $d$ $\mathrm{n}\mathrm{j}$ Boyer C B A hutory of rnathematics Princeton : Princeton Univ Press (All references are to the paperback edition) 1985 $d$ [Couturat 1901] Couturat Louis La logique de Leibniz : apres des documents inidits Paris 1901 (Hildesheim Olms 1985)(rep) [D Alembert 1746] D Alembert Jean le Rood Recherches sur le calcul int\ egral Mdmoires de 1 acadimie de Berlin 1746 (1748) PP $l$ [$\mathrm{d}$alembert 1769] D Alembert Jean le Rond Recherches sur le calcul $\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}^{i}\mathrm{g}\mathrm{r}\mathrm{a}1$ Mimoires de acadimie royale des sciences de pp $Pa\tau\dot{\tau}s1769(1772)$ $Eule\tau\dot{\tau}$ [Euler Opera] Leonhardi Opera Omnia $\mathrm{b}\mathrm{e}\mathrm{r}1\mathrm{i}\mathrm{n}- \mathrm{g}\ddot{\mathrm{o}}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{n}arrow \mathrm{l}\mathrm{e}\mathrm{i}\mathrm{b}\mathrm{z}\mathrm{i}\mathrm{g}- \mathrm{h}\mathrm{e}\mathrm{i}\mathrm{d}\mathrm{e}1\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{g}_{1}$ (1911-) $l$ [Euler 1751] Euler Leonhard Recherches sur ies racines imaginaires des \ equations M\ emoire de acaddmie $\mathrm{v}\mathrm{o}\mathrm{l}$ de Bedin1749(1751) pp ( [Euler Opem] (1) 6 $\mathrm{p}\mathrm{p}$ $78\sim 150$ ) [Euler En-1905] Enestr\"om G Der Briefwechsel zwischen Leonhard Euler und Johan I Bernoulli Bibfiotheca

12 secundi Helmstadt 11816;$ [Gauss 135 Mathematica (3) $6(1905)$ $\mathrm{p}\mathrm{p}16-87$ [Euler Fuss] Comespondance mathimatique et physique de quelques c\ el\ ebres g\ eom\ etres du XVIII si\ ecle 2 vols P H Fuss ed Petersbourg 1843 [Euler Opera Post] Leonhardi Euleri Opera potuma mathernotica et physica 2 vols P H and N Fuss ed Petersbourg 1862 [Fine and Rosenberger 1997] Fine Benjamin : York : Springer 1997) Rosenberger Gerhard The Jundamental theorem of algebra (New [Causs 1799] Gauss Carl Friedrich Demonstratio nova theoremis omnen functionem algebraicam rationalem $\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{s}\mathrm{e} $ $\mathrm{v}\mathrm{e}\mathrm{l}$ integram unius varlabllis in factores reales primi gradus resolvi 1799; [Gauss Werke] vol 3 pp 1-31 [Gauss 1815] Gauss Carl Friedrich Demonstratio nova altera theoremis omnen functionem algebraicam ratiortalem integram unius variabilis in factores reales primi secundi gradus resolvi posse 1815; [Gauss Werke] $\mathrm{v}\mathrm{e}\mathrm{l}$ vol 3 pp [Gauss 1816] Gauss Carl Friedrich Theorematis de resolubilitate functionum algebraicam integrarum in factores reales demonstratio tertia Supplementum commentationis Werke] vol 3 $\mathrm{p}\mathrm{p}$ $\mathrm{p}\mathrm{r}\mathrm{a}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{s}

13 \mathrm{b}\mathrm{a}\mathrm{d}$ $\mathrm{f}\mathrm{r}$ MParmentier Bernoulli $\ovalbox{\tt\small REJECT}$ Fink $[\mathrm{l}\mathrm{m}\mathrm{g}]$ vol $\mathrm{p}\mathrm{p}$ : 1 $3\mathrm{B}$ [Lagrange 1772] Lagrange Joseph Louis Sur la forme des racines imaginaires des \ equations Nouveam $\mathrm{v}\mathrm{o}\mathrm{l}3$ $l$ mirnoiws de Acad\ emie de Berlin 1772 pp ; [Lagrange (Euvoes] $ $ [Leibniz 1702] Specimen novum analyseo pro scientia infini circa summas et quadraturas Acta eruditormrn $\mathrm{p}\mathrm{a}\mathrm{r}\overline{\mathrm{l}}\mathrm{s}$ $(\mathrm{m}\mathrm{a}\mathrm{i} 1702)$ Leibniz$\cdot$ $\mathrm{p}\mathrm{p} $ dans La naissance du calcul diff\ erentiel ( Vrin 1989) $\mathrm{p}\mathrm{p}$ $ $ $\mathrm{j}\mathrm{a}\mathrm{c}$ [Leibniz 1703] Leibniz GW 1920] PP $\prime an $\mathrm{l}\mathrm{e}\mathrm{i}\mathrm{b}\mathrm{n}\mathrm{i}\mathrm{z} $ 3-1 PP $ffl^{-}\mathrm{f}\mathrm{f}\mathrm{i}$ 71-73: [Child [Mahnke 1912] Mahnke D Leibniz auf der Suche nach einer allgemeine Primzahlgleichung $[BM]$ dritte folge (13) $1912arrow 1913$ PP [Neubauer 1978] Neubauer John Syrnbolismus und syrnbotische Logik : die Idee der $ars$ combinatona in der $\mathrm{w}$ Entwicklung der modemen Dichtung (Munchen : 1978) [Novy 1973] Now Lubos Origins of modern algebra ( : Jaroslav Tauer etc) ( Leyden : Noordhoff International Publishing 1973) [Petrova 1974] Petrova Svetlana Sur J histoire des demonstrations analytiques du th\ eor\ eme fondamental de l alg\ ebre Histona mathematica 1 (1974) pp [Serfati 2001] Serfati Michel Math\ ematiques et pensi symbolique chez Leibniz Revue $d$ hestoire des sciences 54/2 annn62001 PP [Struik 1967] Struik Dirk J A Concise History of Mathernatics (3rd rev ed) (New York : 1967) Dover Publications [Struik 1969] Struik Dirk J(ed) A source book in mathematics (Cambridge Mass : Harvard University Press 1969) $\backslash \backslash \#\mathrm{f}\mathrm{l}\ovalbox{\tt\small REJECT}\Re 4$ [ 2001] ffi $\langle$euler L Introductio in analysin infinitorum 2001 $\mathrm{a}$ [ 2003] \hslash a 4 2 \pi $;\mathrm{f}\mathrm{f}\backslash$ 2003 [ 2000] ( 2000 ) [ 1997 ) $\beta$ [ 1999] $\mathrm{j}$ 1997] { At* $\mathrm{f}^{\pm}-\cdot$ 2 $\doteqdot_{\mathrm{d}}^{\pm}$ 3lf $i\backslash $\ovalbox{\tt\small REJECT} \mathrm{f}\mathrm{f}\mathrm{l}$ \mathrm{f}\mathrm{f}\mathrm{i}\{\#$ $\grave{f}\mathrm{f}l\not\in$ ( ( 1999 )