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What is a Galois Group ? ... PDF Print E-mail
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Monday, 21 May 2012 00:00

Évariste Galois

Greetings in the day of the Moon.

         Évariste Galois has been considered one of the Great Minds ever in the development of Mathematics. In fact, many books has been written about his life, the duel that lead him to death.

          What is a Galois Group? ...  What is a Group? ... 

        Before to answer this question I need to introduce another reflection of this Soul. Another person, which name was Niels Heinrik Abel.

(Click to mark)


      Both Europeans, both interact with Cauchy and Abel also speak with Gauss. In fact, Abel present to Carl Friedrich Gauss, and this almost trash the Abel proof.

     The Subjects are Polynomials. And like you know a x + b = c, is a polynomial of first degree, while a x2+b x + c = 0, is a polynomial of second degree and both are solved in easy terms.

     The Polynomial of third degree can be reduced to p3+qx + c = 0, and it needs to solve a polynomial of second degree to offer the real root, which always exist.

      The Polynomial of Fourth degree a x4+b x3 + c x2 + d x + e= 0, can be reduced to two polynomials of second degree.

      For long time in 'know' history ... No one offers answers about 'Over degree 4'. If they can be solved or not.

      In fact Both Young Mathematicians ... solves ... this problem.

      Abel prove that There is NOT POSSIBLE to solve a Polynomial of degree > 4. And Galois ... give a method and a characteristic ... that in mathematics terms ... is a necessary and sufficient condition to solve a Polynomial by radicals.


       The Book,

        never has been published in English (translated to English) and remain in French.

        One of the best ... (but not only books about Galois Theory ... which is a complex concept) can be studied by British, German, French or American books. Just check the Web ...

          I offering these notes written by Yves Laszlo at 'L'Ecole Polytechnique' de Paris.

      These notes has been extended to a book more complete written by David Hernandez.


      My figuring with the books is because you cannot digest easy these concepts at a first glance. You need the books.

      I consider that ANY MATHEMATICIAN need to write a book about Galois Theory ... but

       What is a Galois Group ? ... What is a Group? ...

       A Group is an Abstract Algebraic concept that is defined by an operation. A specific set will be a group if it respect the axioms and properties.

      Galois Theory is based (like the solution of Rubik's Cube) on Groups. Now, what Galois did was like many Mathematicians do, and is to solve the problem ... beginning with the Solution.

       In fact, he claims and ask ... What properties may be the roots of a polynomial? ... The Solutions must be a Group and this Group is called 'Galois Group'.

       You can find examples on the Web. I want to point out a specific polynomial.

x5 + 5 cx + d = 0

       Italian Mathematician ... Gian Battista Malfatti (1731-1807), publish the Malfatti resolvent which is:

(z - c)4 x (z2- 6cz + 25c2) = d4z

       In fact, Malfatti anticipate Galois offering ... one solution ... while remain useful ... it remains incomplete for any case also because it touch degree sixth (6) ... more easy.

        The Galois approach remains definitive because the roots permute ... in their Group.

        You can learn Galois Theory also with this book,


        I want to complete this example on Mathematics ... speaking about the variety in the Mind of Men.

        While French remains the best people for Mathematics, but not the only, Americans remains the best for Economics. British loves ... of course ... Shakespeare and the English Language and Italians have excellence in Physics. Germain are proud in Music ... and rigor.

          What about a Multicultural people ... having the excellent of French in Mathematics, of British in English, of Italians in Physics, of Germain in Music ?

     ... and of course ... this description wants to unite and push Multicultural like a new Political system and don't divide, but Unite.

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.



Giovanni A. Orlando.

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Last Updated on Monday, 21 May 2012 08:22