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Does anyone understand more precisely how to explain the 5th degree equation and elliptic functions accomplishments of Mathematician Graciano Ricalde? I am his great grand-daughter and trying to accurately list his achievements on Wikipedia. I would greatly appreciate your expert guidance. Thank you kindly in advance for any/all responses.

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Do you have any reference to his actual works? The wikipedia article does not have any. Felix Klein showed how to solve the quintic with elliptic functions in the nineteenth century. It is not clear what you are claiming your ancestor did. – Felipe Voloch May 27 '12 at 1:16

Some investigation with Wikipedia and Google books reveals the following.

First of all, as noob explains, there is the mathematical problem that has spanned centuries of solving polynomial equations. The problem has several interpretations, one of which is to find higher-degree versions of the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ for the solutions to the quadratic equation $$ax^2 + bx + c = 0.$$ Such solutions were found for polynomial equations of degree 3 and 4 by Cardano and Ferrari in the 16th century. For polynomial equations of degree 5 and more, it was proven by Abel and Galois that the solution cannot be expressed in terms of radicals (i.e., expressions involving $\sqrt[n]{z}$ for several values of $n$ and $z$). Hermite instead found a solution involving certain functions in advanced calculus that are called elliptic functions. (They generalize trig functions like $\sin z$ and which are indeed related to elliptic curves and originally to the problem of calculating the perimeter of an ellipse.) His solution was simplified and clarified by Kronecker and Klein. Most of this work was before Graciano Ricalde was active as a mathematician.

Enter Ricalde. He corresponded in a French question-and-answer journal called L'Intermédiaire des mathématiciens (which reads a lot like MathOverflow!). In one entry that I think was published in 1898, he asked about solving the quintic equation --- he didn't solve it himself. His question was about reducing the general quintic equation $$ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0$$ to the special form $$y^5 + py + q = 0,$$ which is called Bring-Jerrard form. (In other words, Bring and Jerrard showed how to eliminate three of the terms in the quintic equation.) Obtaining Bring-Jerrard form is a complicated part of the problem, similar to Ferrari's solution to the quartic equation, but the hardest part is to solve the Bring-Jerrard equation; that's the part that requires elliptic functions.

Much later there was a long laudatory biography of Ricalde in a regional publication called Enciclopedia Yucatanense. This biography was cited in Spanish Wikipedia, which apparently led to the English Wikipedia entry. It credited him with solving the quintic equation with elliptic functions, and I think some other major achievements as well. (Although it's a bit hard for me to tell, since I only get a snippet view.) However, in modern mathematical literature, Ricalde is barely cited for anything. The only citation that I found in the arXiv was in a review by Lemmermeyer concerning another algebra problem called Pell's equation. (Lemmermeyer includes Ricalde in a long list of people who solved special cases of Pell's equation, citing the 1901 issue of L'Intermédiaire des mathématiciens.)

My guess is that a local biographer of Ricalde took some topics that Ricalde studied to be Ricalde's own achievements. Moreover, that Ricalde was a competent or aspiring research mathematician in his time, but not a standout.

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I think the solution for the cubic equation was found by Tartaglia. – shahram Sep 16 '13 at 17:31

Given some fixed coefficients $a,b,c,d,e,f$ the fifth degree equation refers to the polynomial of degree $5$, $$ f(x) = a x^5 + b x^4 + c x^3 + d x^2 + e x + f $$

The problem now is to find the exact solutions of the equation

$$ f(x) = 0 $$

in terms of $a,b,c,d,e,f$. A classical question in mathematics asks if it is possible to express these solutions by "radicals", that is, using only the operation of addition, subtraction, multiplicative, division, and taking radicals (i.e square-roots, cube roots, and so on) applied to $a,b,c,d,e,f$.

The answer to this question is known. It is not possible to solve the equation $f(x) = 0$ by radicals when $f$ is a degree 5, or higher, polynomial (as in this case), but possible for polynomials of degree four. This solution comes from a field of mathematics known as Galois theory (with previous work by Abel).

Now enter the elliptic function. The elliptic functions are a class of function with very rich properties and that have been absolutely central in the mathematics of the 19-th century. They still hold an important position to this day.

The relation between the equation $f(x) = 0$ and the elliptic function is, I believe, that while it is impossible to solve the equation $f(x) = 0$ by radicals (i.e in terms of these 5 or 6 operations applied to $a,b,c,d,e,f$), it is possible to express the solutions to the equation $f(x) = 0$ in terms of these 5 or 6 operations and elliptic functions.

I don't know much more beyond this point. If I did say something wrong feel free to correct me.

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OP wants to know how to explain "accomplishments of Mathematician Graciano Ricalde." This does not answer the question, as it gives no indication of what Ricalde had to do with anything. – Gerry Myerson May 26 '12 at 23:24
It's a start. I'm setting the context. – noob May 26 '12 at 23:34
Noob... thank you for setting the context. I am appreciative of all feedback... I hope to learn more precisely about the accomplishments of Mathematician Graciano Ricalde and look forward to hearing more from this community. – Amy May 27 '12 at 2:00

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