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Questions tagged [abstract-algebra]

For questions about the mathematical field abstract algebra that studies algebraic structures, most notably groups, rings and fields.

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How did the terms “center” and “centralizer” come up in group theory?

Usually the word center means the center of a circle. I have encountered the word center in group theory, but do not see any connection with the center of a circle. I think the history of group theory ...
4
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1answer
192 views

What are the modern connections of the Pentagramma Mirificum studied by Gauss?

In the last years, i read a lot about a mathematical object that was discovered by John Napier in 1620 and explored much more deeply by Gauss, who called this "Pentagramma Mirificum" (latin for "the ...
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131 views

Gauss's anticipation of quaternions and their relation to congruences

Recently i read the article "Hamilton, Rodrigues, Gauss, Quaternions and Rotations: A Historical Reassessment", which can be found freely on the internet. This article is by far the most comprehensive ...
2
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1answer
124 views

History of group theory character tables (as used in physics and chemistry)

Does anyone know who started using the symbols A, B, E, T (First column, left) for showing irreducible representations of symmetry groups? In older maths books I see capital gamma. Herein A= totally ...
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Notation $n=efr$ in algebraic number theory

When $\Bbb Q \subset K$ is a field extension of finite degree and when $p \in \Bbb Z$ is a prime number, the ideal $p O_K$ decomposes uniquely as a product $\prod_{i=1}^r P_i^{e_i}$ of prime ideals of ...
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1answer
109 views

Where does the letter S in “$S$-units” and in localization $S^{-1} R$ come from?

In number theory, we may encounter the notion of $S$-unit, $S$-integer, etc. where $S$ is a finite set of prime numbers (for simplicity). For instance, if $S = \{2,3\}$ then the $S$-integers are the ...
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2answers
82 views

What was the significance of Eisenstein's discovery of invariants?

I am trying to decipher a portion of James Joseph Sylvester's 1869 address entitled "The Study That Knows Nothing of Observation", which, among other things, surveys the landscape of 19th century ...
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205 views

(Co)Homology: From topology to the rest of mathematics?

I can appreciate how (co)homology arose in the context of topology/geometry. Trying to get a handle on the handles of spaces leads one to this idea. It's not obvious, but I can see how this would lead ...
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282 views

Whence “homomorphism”, “homomorphic”?

The kernel question leads to another : Today, homomorphism (resp. isomorphism) means what Jordan (1870) had called isomorphism (resp. holoedric isomorphism). How did the switch happen? “Homomorphic” ...
3
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1answer
171 views

Why is the term “kernel” used in algebra? [duplicate]

What was the motivation to use the word "kernel" in algebra to denote the set of all arguments which are mapped to the idendity element (by a homomorphism)? Who introduced it?
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What color were Emmy Noether's eyes? [closed]

I have reposted this question from MathOverflow because it is questionably relevant there. I am looking for a reference that definitively says what the color of Emmy Noether's eyes were. More ...
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1answer
81 views

Where did the notion of the product in a category first appear?

In his book Category Theory [1] Awodey writes the following about it: Next, we are going to see the categorical definition of a product of two objects in a category. This was first given by Mac ...
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1answer
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Who first identified the group structure of an elliptic curve?

I find it amazing that the geometric construction that underlies the group law for elliptic curves gives rise to a group law. Q: Who was the first to identify the group law for elliptic curves and, ...
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1answer
218 views

Abbreviated Notation for Groups, Rings and Fields

Groups, Rings and Fields are often referred to by the set involved without mention of the operation(s). For example, the "group (G,+)" may be called the "group G". When did this practice originate ...
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1answer
317 views

Who was first to differentiate between prime and irreducible elements?

I recently learned about irreducible and prime elements in a commutative ring. However, my professor was not quite sure who was the first to make this distinction, or give an example of an irreducible ...
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322 views

What is the origin of “an algebra” as in vector space with multiplication?

What is the origin of calling a vector space over a field $F$ endowed with multiplication an algebra? Tried searching, but not surprisingly Google likes to drop the article and just bring me to the ...
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1answer
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Sum of like powers in real numbers

When and who was the first person to discover a correct formula for the real number $r$ in terms of any given three positive distinct integers $x<y<z$ such that $$x^r + y^r = z^r\,.$$ The ...
7
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1answer
261 views

Who discovered the topological proof of Nielsen-Schreier theorem?

The celebrated Nielsen-Schreier theorem in group theory says subgroup of a free group is free. This was proved for finitely generated subgroups of free groups by Jakob Nielsen in 1921, which involved ...
3
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1answer
150 views

Jordan called isomorphisms (iso.) and homomorphisms “iso. holoedriques” and “iso. meriedriques” respectively; translation of holoe/meried-driques?

Stillwell mentions in his Elements of Algebra: The first to use the term "isomorphism" was Jordan, in his Traite des Substitutions [1870], the first textbook on group theory...Jordan used the word "...
7
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2answers
169 views

Where I can find the translated manuscript of Abel?

I am looking for the translated manuscript of Abel where he proved the unsolvability of the quintic. Can anyone give me a pointer? I tried google but nothing came up.
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2answers
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On the notion of a chain (as for example in chain complex)

The thing with mathematics is that on one if you define something, you are completely free in choosing any name you want, and on the other hand you should find a meaningful name that evokes some ...
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When did people know that all real polynomials of degree greater than 2 are reducible?

Admittedly, this may not be a research level question, but I am deeply curious about this. Let $f(x) \in \mathbb{R}[x]$, and write $d = \deg f$. It is well known that if $\deg f > 2$, then $f$ is ...
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1answer
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From the perspective of etymology, why was the word “magma” chosen to describe a set with a single binary operation defined on it?

According to Wikipedia, the choice of vocabulary was made partially to avoid overloading the term "groupoid". However, that still does not explain etymologically speaking, "magma" was chosen instead ...
3
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1answer
165 views

Why did Kronecker develop the “adjoining a root” construction?

Kronecker is generally credited with the formalization of "adjoining a root to $f(x)=0$". Nowadays it is interpreted as the quotient $K[x]/(f)$, where $K$ is some appropriate algebraic structure in a ...
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History of the Wreath product

Why is the wreath product so named? If possible, please provide a citation.
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1answer
137 views

What were the initial applications of finite fields?

Finite fields, I believe, were introduced by Galois in his paper "Sur la théorie des nombres", found on page 398 of his Oeuvres. In this paper Galois introduces the idea of taking a polynomial ...
5
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1answer
123 views

What's the origin and meaning of the adjective “free” in mathematics?

It's pretty common to call a group, ring or module free when it has a 'basis', but unlike other mathematical definitions whose names can be easily related to the concept they describe (e.g. the ...
7
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1answer
180 views

How did the 'Poincaré patches' get their name?

De Sitter space and Anti de Sitter space are two of the most important solutions to the Einstein field equations. One famous method to obtain these spacetimes is to consider a $N$-dimensional ...
13
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1answer
173 views

When was the modern field theory approach to Galois theory developed?

It is well known that Galois, and other mathematicians around that time, considered Galois groups to be permutation groups and approached Galois theory in this manner. At some point the theory took a ...