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9

The systematic modern terminology and presentation of the Galois theory is due to Artin, a part of his joint project with Emmy Noether to reformulate the "concrete" older algebra in abstract terms, inspired by Dedekind. It was Artin who finally detached the Galois theory from the problem of solving equations algebraically, and gave a presentation that freely ...


8

In 1969 Lang wrote an article for the Columbia Daily Spectator, Don't Blame Us if You Flunk Math (Volume III, Number 4, December 8, 1969). The phrasing of the subline illustrates how much the times have changed:"A fifteen minute quiz raises questions what kind of people should be taught what kinds of math at Columbia". He provides a basic introductory test ...


7

Bombelli did not have a general method for such transformation, nor does it exist. Doing so is equivalent to solving the irreducible case of the cubic, which is impossible in (real) radicals. It is solvable in inverse trigonometric functions, as Vieta showed later, but simplifying his trigonometric expression is itself a chore. However, when everything is ...


6

It is in the third book of La Géometrie: I could also add rules for equations of the fifth, sixth, and higher degrees, but I prefer to consider them all together and to state the following general rule : ...and, consequently, if it is of the third or fourth degree, the problem depending upon it is solid; if of the fifth or sixth, the ...


6

There is a problem with how the question is phrased, which is unfortunately common. Modern concepts pull together many different threads which come apart when projected into the past far enough. "Cubic equation" and "real numbers" are two such concepts. When they are used to ask questions about discoveries before 19th, or at least 17th, century there can be ...


5

This question comes up often, but there is no up to date scholarly study of it that I know of. The most comprehensive recent accounts (and they are rather brief) seem to be Jeff Miller's Earliest Uses of Symbols of Operation and Peterson's post on Math Forum, parts of which are aped on various pop-sites, often without attribution. But Peterson admits:"I have ...


5

For sure, the technique was used by Leonhard Euler (15 April 1707 – 18 September 1783). See : Elements of Algebra (English transl., 1822), page 208: Since the two equations are, $x+y=a$, and $x-y=b$; if we add the one to the other, we have $2x=a+b$. The first German edition of the Elements was in 1770. An earlier example ...


4

It is easy to find fault with Descartes and Leibniz spent his life doing it (see Belaval Y., Lz critique de Desc., P.1960). Descartes knew that some problems of higher degrees are reductible and erroneously believed that it is the general case. The question here however concerns a paraphrase without reference and asks for a good match. such as e.g. La ...


3

Nothing new was discovered after Boyer published his textbook. In 2014, Victor J.Katz and Karen Parshall published their book Taming the Unknown: A history of algebra from antiquity to the early twentieth century, in which they write (chapter 7): While influencial, al-Khwārizmī's was evidently not the ony algebra text of the time. A fragment of a ...


2

My impression is that Artin's role in the development of Galois theory is usually greatly overrated. Kiernan's article jumps from 1900 to Artin in the late 1930s. Certainly a "modern" Galois theory existed during the intervening years --- see for example, Albert's Algebra books. Artin's main contribution was to give a proof of the main theorem of Galois ...


2

but for how long has PEMDAS been widely taught in high school mathematics classes? I assume this will differ for different parts of the world, so please include what countries or regions you can speak for. The first appearance of an explicit PEMDAS rule in Dutch is in an appendix of a textbook on algebra for the military academy (1838), aimed at military ...


1

There most certainly were multiple 'inventors' -- or 'discoverers,' depending on your phiilosophy, of basic algebra. Even today, there are youngsters who develop algebra independently. As to who should get credit for bringing a given chunk of math to the world, that's not really an answerable question. The growth of knowledge is nonmonotonic; ...


1

According to an excerpt from The Heritage of Thales By W.S. Anglin, J. Lambek, via GoogleBooks, a method was found: Note that this doesn't indicate a proof that there had to be a real root. The Wikipedia page on cubic equations adds that Cardano noticed that Tartaglia's method sometimes required him to extract the square root of a negative number. He ...


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