The author's main idea can be summarized as follows: first, find a root of the univariate quintic equation x 1, and then simplify the univariate quintic equation into a univariate quartic equation, so the problem is simple. After all, the solution of univariate quartic equation is a solved problem. The key problem is how to solve x 1! The author obtains x 1 by using the decomposition coefficient method and investigating the range of x 1.
In the preface, the author said, "Whether all the quintic equations and each example we have solved here have been verified to determine whether they are correct or not, that is to say, it is a foregone conclusion to test whether they are correct or not, and there is no deviation, difference, error or omission, and there is no authoritative problem." Although there are only 180 pages in the book, these 180 pages are almost all calculations. This kind of work is pioneering in China, by solving the quintic equation of 16 one by one, supplemented by verification. The author's spirit of hard study and exploration is really worth learning! But the question is: does this mean "cracking" the quintic equation of one yuan?
As mentioned in detail earlier, no matter Abel or Galois, what they want to prove is that there is no radical solution to the general equations of quintic and above, that is to say, algebraic equations of quintic and above cannot have a finite number of addition, subtraction, multiplication, division or square root operations to get all the solutions of the equations, like quadratic, cubic and quartic equations. That is, Abel theorem points out that when n ≥5, the general polynomial of degree n cannot be solved by root sign. Although this refers to general polynomials, even if the coefficients are integers, polynomials of degree 4 or more may not be solved by root signs. Both Abel and Galois tried to solve the equation with roots.
x？ 4x +2 is a typical example. Of course, the roots of some special univariate quintic equations can still be solved by other methods. Of course, if human beings are always limited to using certain methods and tools, then mathematics cannot develop. If we are not limited to the operation that only uses the complex number field and the root of the root, then there is probably a way to solve the root of the quintic equation with one variable. In fact, we can find the real root of any polynomial f (x )∈ R [x] by Newton method: if R is the real root of f (x) and h0 is a "good" approximation of R, then r = lim hn, where hn+1 = hn? F (HN) f ′ (HN). In addition, there is Hermite method to find the roots of quintic equation by using elliptic n→∞ modular function. Kronecker mentioned in a letter to Hermite and a later article that the general quintic equation can be solved by elliptic modulus function.
Generally speaking, there is no radical solution to the quintic equation of one yuan. However, the absence of a radical solution does not mean that it cannot be solved at all. If we can find the solution or approximate solution of the univariate quintic equation, then obviously we can simplify the univariate quintic equation to the univariate quartic equation, so the equation is solvable. It is along this line that the author of this book solved the quintic equation of one yuan. It is a common idea in mathematical research to transform complex problems into relatively simple problems and transform high-order equations into low-order equations. Obviously, this book does not mean that Galois's theory is out of date. Galois theory proves that the quintic equation of one variable has no root solution, that is, there is no formula for finding the root. There is no unified formula for solving the quintic equation of one variable in this book. Instead, the coefficients of the quintic equation are classified and solved in different ways. And in the process of solving, approximate solutions are sometimes used. In the preface of the book, the author said: "Galois has specially proposed and proved: a quintic equation of one yuan, such as X? X+ 1 =0 cannot be solved by the root formula, but the author has solved it because this problem only needs approximate calculation. " In this respect, we beg to differ. What Galois means is that X can't be solved by the root formula? X+ 1 =0, and the author of this book admits that their "solution" is the result of "using approximate calculation method". After carefully reading the solution of this book, it is not difficult to find that the author is using a lot of calculations to find the approximate solution x 1 and then solve a quartic equation. This can't be confused with Galois theory. The author's so-called "cracking" theory can easily give people an X? This galois doesn't even
The solvable equation was solved by the author. If only the root of the equation is found, the author's "crack" is established; But Galois's theory is the best answer to the old problem of finding the root formula of the general quintic equation with one yuan.