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I am a physicist whose grasp of "proper" mathematics is probably quite lacking. I have some books on mathematical physics, but when more pure math is involved, I do not know where to look. In particular, I've been using Wikipedia to figure out how to solve cubic and quartic equations, and now I am looking for a proper reference book. I am unsure what specific branch of maths the solution of such equations pertain to (algebra something I assume). The book I am looking for should be, if possible, a rather standard reference to mathematicians. It should contain a description of Cardano's method for the solution of depressed cubics as well as Descartes' method for the solution of quartics. I did manage to find Uspensky's Theory of Equations, but there is no mention of Descartes' method.

J. W. Tanner
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BitterDecoction
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    Wikipedia gives references. See also NIST. https://dlmf.nist.gov/1.11#iii –  Oct 30 '24 at 18:01
  • Where in physics do you need exact radical solutions for cubic and quartic polynomials? I took quite a bit of physics in the late 1970s to early 1980s (upper level undergraduate and beginning graduate level) and this never came up. That said, look for Theory of Equations books (a standard undergraduate level topic taught up until roughly the mid 1950s) – Dave L. Renfro Oct 30 '24 at 19:09
  • Yes, but I have found Wikipedia's references to be unhelpful. The only citations on Descartes' method are Descartes' own work, which is not what I am looking for, and a 53 EUR paper. The only reference on Cardano's formula on Wikipedia is a webpage. As I mentioned in my question, I am looking for a book, a standard reference for mathematicians. – BitterDecoction Oct 30 '24 at 19:11
  • See: books on polynomial algebra with much detail than pre-calc books AND Where do students learn to solve polynomial equations these days? (Mathematics Educators SE) – Dave L. Renfro Oct 30 '24 at 19:16
  • I wrote up the case $x^3 + px + q = 0,$ detail for when $p >0$ where there is one real root..... https://math.stackexchange.com/questions/4987496/dubious-step-while-solving-8k2k-5/4987512#4987512 – Will Jagy Oct 30 '24 at 19:16
  • @DaveL.Renfro, I never came across such equations as an undergrad. But it really depends on the model you are looking at. In my case it comes from a condition I put on a simplified 1D model. It appears the book you shared is not unique in the sense that many different authors wrote books with the same name. Which author are you referring to? Incidentally I mentioned in my question the one written by Uspensky. – BitterDecoction Oct 30 '24 at 19:27
  • There is no standard reference for the theory of equations, no more than there is a standard reference for calculus, for linear algebra, for fluid dynamics, etc. However, the most complete treatment I can think of off-hand is The Theory of Equations by William Snow Burnside and Arthur William Panton. Regarding Descartes method in particular, see the 3rd paragraph AND the section titled "3. Quadratic Factors of $;x^4 + 10x^2 - ; 96x - 71$" in this MSE answer and the references given there. – Dave L. Renfro Oct 30 '24 at 19:34
  • It appears the book you shared is not unique in the sense that many different authors wrote books with the same name. --- I said "look for Theory of Equations books", analogous to saying (for example) "look for Calculus books or "look for Linear Algebra books" or "look for Fluid Dynamics books. Maybe I shouldn't have italicized and capitalized "Theory of Equations" $\ldots$ – Dave L. Renfro Oct 30 '24 at 19:41
  • My bad. There is such a thing as standard references. In fluid dynamics you have Batchelor, Landau & Lifshitz, Tritton (for physicists at least). In classical electrodynamics the two big names are Jackson and Griffith. In mechanics the standard reference is Goldstein. I can go on. This is nothing new. I will take a look at Burnside & Panton though. In one of the links I also found Dickson which I assume is one of the standard references as it is one of the two references linked on the Theory of Equations Wikipedia page. – BitterDecoction Oct 30 '24 at 20:02
  • My bad also. I was too busy looking up links and such that I didn't read carefully enough. You wrote "a rather standard reference", not "the standard reference". (I gave some standard physics text references when I was dealing a lot with physics in this MSE answer.) In addition to Uspensky, Dickson, Burnside/Panton I'd add An Introduction To The Modern Theory Of Equations by Cajori (1904). I know of many others, but these are probably the best known. Incidentally, there are 3 different but similar Dickson books (1914, 1922, 1939). – Dave L. Renfro Oct 30 '24 at 20:14
  • @njuffa: Neumark's book (which I happen to have a hardback copy of) might not be what the OP is interested in. It deals with some numerical methods and associated numerical tables for the approximate solution of various special types of cubic equations. (from the Preface) This book suggests a rapid and efficient method of computing the roots of an arbitrary cubic equation with real coefficients, by using specially computed 5-figure tables, and also a method of factorizing an arbitrary quartic equation by an appropriate use of a resolvent cubic. – Dave L. Renfro Oct 31 '24 at 01:44

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