Since the establishment of "Scientific Space," the "Problem Encyclopedia" category has been set up, but the content has always been sparse, mainly because I have been too lazy to summarize problems. Now I need to develop the habit of being good at thinking and summarizing.
A few days ago, I went online to print and read Celestial Encounters and The Equation That Couldn’t Be Solved; both are books I am very interested in. I remember back in junior high school when reading Selected Lectures on the History of Mathematics, my greatest interest was in solving equations (radical solutions). Through research, I understood the root-finding formulas for equations of degrees 1 to 4, and through reading, I learned that algebraic equations of degree higher than 4 have no general radical solution. What a joyful thing that was at the time!!
Now, after briefly reading The Equation That Couldn’t Be Solved and combining it with some of my previous summaries on algebraic equations, I propose a question:
If the roots of an arbitrary n-th degree equation in one variable \sum_{i=0}^{n} a_i x^i=0 are denoted as x_i=R_{n,i}(a_0,a_1,...,a_n),
Then, does there exist an n > 3 such that the roots of any (n+1)-th degree equation in one variable can be calculated through finite steps of addition, subtraction, multiplication, division, powers, roots, and R_{j,i} (where j can be any integer from 1 to n)?
This question can be rephrased in an approximate but non-equivalent way:
If equations of degrees 1, 2, ..., n can all be solved by radicals, can an (n+1)-th degree equation then have a radical solution?
In other words, can the roots of an (n+1)-th degree equation be expressed as a finite number of operations involving addition, subtraction, multiplication, division, powers, roots, and the roots of equations of degrees 1 to n?
(Regardless of the correctness of the premise, clearly n=4 no longer holds; but is it possible for n=5, 6, 7, 8, \dots, etc.?)
Looking forward to someone being able to solve this ^_^
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