According to the ancient Egyptian cursive script, as early as around 1700 BC, it was found that when a≠0, ax = b had roots. With the passage of time and the development of mathematics, by the centuries BC, the Babylonians actually knew that (when a≠0) had roots.
At that time, people only realized that the so-called positive real root is a root, and the concepts and theories of zero, negative number, irrational number and complex number were not recognized and gradually improved until the sixteenth to eighteenth centuries. According to the Babylonian literature, the quadratic equation was solved at that time:
The answer is:
This urges people to think further, can this formula be found for any number of equations? It took more than 2000 years to find the root formula of cubic equation, and it was not until16th century European Renaissance that several Italian mathematicians found it. This is the so-called Cardan (150 1- 1576) formula, and its original idea is: in.
After the variables are replaced, the equation is transformed into
( 1)
It no longer contains the square term. Suppose m and n are two undetermined numbers, then there is.
If m is taken, n is satisfied.
The corresponding y value must satisfy the formula (1). On the other hand, by
free
So, when taking it,
When and make, find a root of the original cubic equation.
Its other two roots are
There are two roots that are not 1
There is an interesting story in the invention of the root formula of cubic equation. More than 400 years ago, there was a math contest in Italy. One of the contestants is Fio (1the first half of the 6th century), who is a student of Ferro (1465- 1526), president of the Mathematical Society of Bologna, Italy. On the other hand, Tarta Ria (1500- 1557), a professor of mathematics in Venice, stuttered after being injured while doing Colla. Like Ferro, Tarta Ria once solved another one at 1530. This aroused the dissatisfaction of the Philippines and Russia, and held an open competition in Milan Cathedral on February 22nd, 1535. Thirty cubic equations are drawn on each side. As a result, Tarta was solved in two hours, while the Philippines and Russia handed in a blank sheet of paper. After 154 1, Tarta obtained the general solution of cubic equation, and prepared to publish his solution in a book after translating the works of Euclid and Archimedes. At this point, Cardin appeared. He repeatedly begged Tarta to give him a poem with obscure sentences. This poem is not well written, but it does contain every step of the solution. He himself said, "I don't mind if there is no good sentence in this poem." In order to remember this rule, you can use this poem as a tool. " After getting all this, Cardin was treacherous. 1545, he published this solution in Dafa, and came to the conclusion that Tarta's method is Ferro's method, which was learned in the competition with the Philippines and Russia. This led to Tarta's great anger and declared war on Kadan. Both sides set 3 1 questions, and limited 15 to hand in papers. Cardin sent his student Ferrari (Ferrari, 1522- 1565) to accept the challenge. As a result, Tarta worked out most of the questions in seven days, while Ferrari only handed in the papers for five months and only got one question right. Tarta Leah wanted to finish a masterpiece including his new algorithm, but unfortunately, he died before his ambition was rewarded. Shortly after the cubic equation was solved, Ferrari, Cardin's servant and student, got the solution of the quartic equation. The main idea is: for quartic equation
(2)
The parameter t is introduced and expressed as follows
(3)
It is easy to verify that (2) and (3) are the same. In order to ensure that the right side of equation (3) is completely square, its discriminant can be 0:
T is a cubic equation.
Any one of them. Let this root be the t value in (3).
Move right and left, and decompose the factor to get two quadratic equations.
In this way, the root of the quartic equation is the root of a cubic equation and two quadratic equations, and it is considered that the problem of solving the quartic equation is also solved. With this breakthrough, mathematicians devote themselves to finding the solution of quintic equation with great interest and confidence. They found that for an equation with no more than four degrees, a formula for calculating roots can be obtained, and each root can be expressed by adding, subtracting, multiplying, dividing and squaring the coefficients of the original equation. We simply call this matter solvable by the root sign, so people assert that there must be such a formula for finding the root of the quintic equation. In this respect, some famous mathematicians at that time, such as Euler (1707- 1783), Vandermonde, (1735- 1796) and Lagrange (1796). 1765- 1822) and gauss (1777- 1855) were convinced, so they tried their best to find it, but they all ended in failure.
First of all, Lagrange doubts the existence of this formula. He deeply analyzed the methods of solving algebraic equations below quintic obtained by predecessors, and found that all the auxiliary equations with lower degree can be replaced by appropriate variables (later called Lagrange resolvent), but the number of auxiliary equations obtained by this method for quintic equations rose to six, so this road was blocked! 177 1 year, Lagrange published a long article "Thoughts on Algebraic Solution of Equations" and raised this question. By 18 13, his disciple, Italian doctor Rufini, finally proved that Lagrange's method of finding the resolvent was really invalid for the quintic equation. As early as 180 1, Gauss also realized that this problem might not be solved. But there is no proof of "non-existence", including Lagrange.
The first person to prove that "equations with quartic degree or more cannot be solved by radical sign" is Abel (Abdl, 1802- 1829), a young Norwegian mathematician. He read the works of Lagrange and Gauss on equation theory in middle school, and discussed the solution of higher-order equations. In 1824- 1826, but after reading it, Gauss said, "It's terrible to write this kind of thing", which means he didn't understand it. Abel had many unique achievements in mathematics, which were not taken seriously at that time. Due to poverty and illness, he died of tuberculosis on April 6, 2008, at the age of 27. Shortly before his death, he told Legendre (1752- 1833) some research results. Just three days after his death, Berlin sent him a letter of appointment as a professor.
But Rufini and Abel's proof is not very clear after all, and there are even some loopholes. Abel did not give a criterion to judge whether a higher-order algebraic equation with a specific numerical coefficient can be solved by root sign. As a history, their achievements are undeniable, but compared with the brilliant achievements of Galois that soon appeared, they are greatly inferior!
Galois (1811-1832) is a young French mathematician. /kloc-entered a famous public middle school in Paris at the age of 0/5, preferring mathematics. Later, I wanted to enter the engineering university, but I failed twice, and I only entered a preparatory course. At this time, he specializes in algebraic solution of quintic equation. In the first year, he wrote Fourier's article. When Galois was 1828 and 17 years old, he wrote two papers, such as Algebraic Solutions of Quintic Equations, which were sent to the French Academy of Sciences, but were lost by Cauchy (1789- 1875). Later, he put another article. Soon, Foley died, and the matter went away. 183 1 in, Galois completed the article "Solvability Conditions for Solving Equations by Roots", but the review opinion of Academician Poisson (178 1- 1840) was "completely incomprehensible" and was rejected. Unfortunately, Galois died on May 3 1, 1832, when he was less than 2 1 year old. On the eve of the duel, he knew it was meaningless to die for his girlfriend, but he was not to be outdone. That night, he was so nervous and nervous that he even shouted "I don't have time!" " In desperation, he scribbled a few pages about his findings in equation theory and sent them to his friends, with the following sentence: "You can openly ask Jacoby or Gauss not to comment on the truth of these theorems, but on their importance. I hope someone will find this pile of things useful to them in the future. " After 14 years, 1864, joseph Liouville (liouville, 1809- 1882) published some articles in his magazine Pure and Applied Mathematics. Jordan (1838- 1892) introduced Galois' theory comprehensively and clearly for the first time in his book Monographs on Permutation and Algebraic Equations published in 1870. This way. Galois's transcendental genius thought is gradually understood and recognized by people, and has become a booming discipline-abstract algebra. Galois avoided Lagrange's elusive resolvent and skillfully used the tool of permutation group. He not only proved the following general algebraic equations:
When n≥5, it is impossible to find the root with the root sign. In addition, the criterion that the algebraic equation of a specific mathematical coefficient can be solved by the root number is established, and an example of the algebraic equation of a digital coefficient that cannot be solved by the root number is given. In this way, he completely solved the problem that puzzled many mathematicians for more than 200 years. Not only that, galois also discovered. His strange ideas and ingenious methods have now become the central content of all algebra. At this point, he is well-deserved as one of the creators of abstract algebra. His contribution is not limited to solving the problem of solving the roots of algebraic equations.
With the passage of time, Galois's outstanding contribution has been recognized by mathematicians more and more. His academic thoughts had a far-reaching impact on modern mathematics: the group theory he initiated gradually penetrated into other branches of mathematics, as well as crystallography, theoretical physics and other fields, providing powerful mathematical tools for these fields. For example, group theory proves that there are only 230 kinds of crystals, which provides a unified method for the study of equation roots, crystal structure, spatial transformation, symmetry of elementary particles and so on. By the 20th century, the concept of group theory played an important role in the whole mathematics and became one of the foundations of modern mathematics.