Contributions to mathematics Évariste Galois

1 contributions mathematics

1.1 algebra
1.2 galois theory
1.3 analysis
1.4 continued fractions

contributions mathematics

the final page of galois mathematical testament, in own hand. phrase decipher mess ( déchiffrer tout ce gâchis ) on second last line.

from closing lines of letter galois friend auguste chevalier, dated may 29, 1832, 2 days before galois death:

tu prieras publiquement jacobi ou gauss de donner leur avis, non sur la vérité, mais sur l importance des théorèmes.

après cela, il y aura, j espère, des gens qui trouveront leur profit à déchiffrer tout ce gâchis.

(ask jacobi or gauss publicly give opinion, not truth, importance of these theorems. later there be, hope, people find advantage decipher mess.)

within 60 or pages of galois collected works many important ideas have had far-reaching consequences branches of mathematics. work has been compared of niels henrik abel, mathematician died @ young age, , of work had significant overlap.

algebra

while many mathematicians before galois gave consideration known groups, galois first use word group (in french groupe) in sense close technical sense understood today, making him among founders of branch of algebra known group theory. developed concept today known normal subgroup. called decomposition of group left , right cosets proper decomposition if left , right cosets coincide, today known normal subgroup. introduced concept of finite field (also known galois field in honor), in same form understood today.

in last letter chevalier , attached manuscripts, second of three, made basic studies of linear groups on finite fields:

he constructed general linear group on prime field, gl(ν, p) , computed order, in studying galois group of general equation of degree p.
he constructed projective special linear group psl(2,p). galois constructed them fractional linear transforms, , observed simple except if p 2 or 3. these second family of finite simple groups, after alternating groups.
he noted exceptional fact psl(2,p) simple , acts on p points if , if p 5, 7, or 11.

galois theory

galois significant contribution mathematics development of galois theory. realized algebraic solution polynomial equation related structure of group of permutations associated roots of polynomial, galois group of polynomial. found equation solved in radicals if 1 can find series of subgroups of galois group, each 1 normal in successor abelian quotient, or galois group solvable. proved fertile approach, later mathematicians adapted many other fields of mathematics besides theory of equations galois applied it.

analysis

galois made contributions theory of abelian integrals , continued fractions.

as written in last letter, galois passed study of elliptic functions consideration of integrals of general algebraic differentials, today called abelian integrals. classified these integrals 3 categories.

continued fractions

in first paper in 1828, galois proved regular continued fraction represents quadratic surd ζ purely periodic if , if ζ reduced surd, is,



ζ
>
1


{\displaystyle \zeta >1}

, conjugate



η


{\displaystyle \eta }

satisfies



−
1
<
η
<
0


{\displaystyle -1<\eta <0}

.

in fact, galois showed more this. proved if ζ reduced quadratic surd , η conjugate, continued fractions ζ , (−1/η) both purely periodic, , repeating block in 1 of continued fractions mirror image of repeating block in other. in symbols have

ζ



=
[




a

0


;

a

1


,

a

2


,
…
,

a

m
−
1



¯


]







−
1

η





=
[




a

m
−
1


;

a

m
−
2


,

a

m
−
3


,
…
,

a

0



¯


]







{\displaystyle {\begin{aligned}\zeta &=[{\overline {a_{0};a_{1},a_{2},\dots ,a_{m-1}}}]\\[3pt]{\frac {-1}{\eta }}&=[{\overline {a_{m-1};a_{m-2},a_{m-3},\dots ,a_{0}}}]\,\end{aligned}}}

where ζ reduced quadratic surd, , η conjugate.

from these 2 theorems of galois result known lagrange can deduced. if r > 1 rational number not perfect square, then

r


=
[

a

0


;




a

1


,

a

2


,
…
,

a

2


,

a

1


,
2

a

0



¯


]
.



{\displaystyle {\sqrt {r}}=[a_{0};{\overline {a_{1},a_{2},\dots ,a_{2},a_{1},2a_{0}}}].\,}

in particular, if n non-square positive integer, regular continued fraction expansion of √n contains repeating block of length m, in first m − 1 partial denominators form palindromic string.