Are there formulas for solving first-degree algebraic equations? Equations like 2x - 6 = 0, x2 - 3x + 2 = 0, x3 - 5x - 12 = 0, etc. are all called first-degree algebraic equations. According to the highest degree of the term, they are respectively called first-degree linear equations, first-degree quadratic equations, first-degree cubic equations, etc. For first-degree linear equations, the solution method is extremely simple. As for the solution of first-degree quadratic algebraic equations, many ancient civilizations also obtained correct results through exploration. In fact, for the general first-degree quadratic equation ax2 + bx + c = 0 (a ≠ 0), we have the following solution formula: In this formula, the solution of the equation is expressed through the addition, subtraction, multiplication, and radical operations of the coefficients of the equation. Therefore, the solution given by this formula is called the radical solution of the equation.
European mathematicians mastered the method of solving quadratic equations in the 12th century. After that, they began to explore the radical solution of first-degree cubic equations. Over the centuries, many people's efforts ended in failure. In 1494, some mathematicians even made pessimistic predictions that solving cubic equations was as impossible as squaring the circle. However, around 1500 AD, the Italian mathematician Ferrero initially succeeded in obtaining a solution formula for missing-term first-degree cubic equations such as x3 + mx = n. However, he did not publish his results and only passed on his solution to his students Fiore and others before his death. Soon after, another Italian mathematician, Tartaglia, claimed to have obtained a solution method for cubic equations. In 1535, a public competition took place between Fiore and Tartaglia. The two agreed to compete in Milan, with each side presenting 30 cubic equations. It was agreed that whoever solved more questions would win. Initially, Tartaglia only mastered the solution of first-degree cubic equations without a constant term, such as x3 + mx2. In order to win the competition, he spent many days racking his brain and finally found solutions for various types of first-degree cubic equations on the eve of the competition. The result of the competition was that Tartaglia achieved a brilliant victory.
Shortly after the competition, another mathematician, Cardano (also known as "Cardan"), heard about this. Thereafter, he repeatedly asked Tartaglia for the solution of cubic equations. Finally, after swearing never to divulge the secret, Tartaglia revealed his secret to Cardano. After that, Cardano, through his research, finally obtained solutions and proofs for all types of cubic equations. In 1545, Cardano published his most important mathematical work, "The Ars Magna", and made his discoveries public. At the same time, the book also included an important discovery of his student Ferrari: the general solution of first-degree quartic equations. Cardano's breach of trust angered Tartaglia, and an argument inevitably ensued. On August 10, 1548, a public debate in Milan brought the conflict to a head. In this public but unfair competition with Ferrari, Tartaglia failed. Until now, the solution formula for cubic equations is still called the Cardano formula.
However, mathematics never stops. The solution of one problem often means the birth of new problems. After "The Ars Magna", people immediately embarked on the next competition, that is, how to find the root formula for fifth-degree equations. No one doubted that as long as there were enough talents, first-degree quintic equations should also be solvable. But a century passed, and another century passed, and no one succeeded.
In late 1823, a real breakthrough occurred. The 21-year-old Norwegian mathematician Abel proved that there is no radical solution for general first-degree quintic algebraic equations. This is precisely the reason why people's search for this "Holy Grail" had failed over the past two centuries. AbelYou copied several copies of your research results. To save printing costs, he condensed the core of his ideas into just six pages. The overly simplified proof made his paper difficult to understand, so it failed to impress the mathematical community at the time. Abel, who was tormented by poverty and illness throughout his life, died of illness on April 6, 1829, before he turned 27.
Shortly after Abel's death, his proof began to be accepted, but there were still new challenges to be solved. Which equations could be solved by radicals, which could not, and was there a suitable method of discrimination? Abel's premature death prevented him from solving this problem, so it was left to Galois, one of the most legendary and tragic mathematicians in the history of mathematics.
At the age of 17, Galois had already made progress in this area, and he submitted several research papers over the next few years. However, his papers were either returned, lost, or misunderstood, and Galois, who had not been recognized by the mathematical community, became increasingly frustrated. Subsequently, a series of unfortunate personal events also followed. Under heavy blows, his enthusiasm turned to politics, for which he was imprisoned twice. In 1832, he got involved in a meaningless duel. The night before the duel, he worked all night and gave a brief description of his research results. On May 30, 1832, Galois fell in the duel and died the next day, at the age of 21. More than 10 years later, his papers were re-edited and published, and his genius was recognized.
The beauty of Galois's ideas lies in the fact that he not only proved that general univariate quintic (and higher) algebraic equations cannot be solved by radicals, but also explained why general univariate quadratic, cubic, and quartic equations can be solved. His research also provided a clear criterion for whether algebraic equations have radical solutions, thus completing the 300-year puzzle of the solvability of algebraic equations by radicals. The epoch-making significance of Galois's work lies in the fact that in the process of solving this problem, he introduced the concept of group theory and opened up a new era of mathematical research.