5
votes
2answers
442 views

What fields (and operators acting on those fields) might form the basis of alien mathematics?

Addition and multiplication, according to most histories, arose in human civilizations out of a need to count a finite number of objects, and then later on especially, to measure land. What might be ...
7
votes
2answers
216 views

Origin of the modern definition of the tensor product

Due to whom is the modern (i.e. via its universal property) definition of the tensor product, and in which article was it communicated?
3
votes
1answer
102 views

unique factorisation fails for cyclotomic integers $p>23$

Background: I have stopped doing algebra a long time ago and I am not that interested in the nitty-gritty details of proofs, but I am interested in maths history. ...
3
votes
1answer
65 views

Hamilton's three dimensional algebra

The popular story of the discovery of the quaternions goes very roughly as follows. William Rowan Hamilton has interested in the construction of an algebra of triplets that would in some ways be ...
0
votes
1answer
155 views

Definition: finite type vs finitely generated

The mathematical term "finite type" appears more and more in the modern articles nowadays. But it is still hard to be found in the standard textbooks. I learned the definition of it from Stacks ...
11
votes
2answers
269 views

Before Abel's proof, what did they used for trying to find the general solution for quintics?

Whenever I read about the history of algebra, I end up with the same conclusion: They solved the general cubic, then the general quartic and then spent lots of years trying to solve the general ...
3
votes
1answer
63 views

Diophantus math

Find two numbers such that their difference and also the difference of their cubes are given numbers; say, their difference is 6 and the difference of their cubes are 504. Call the numbers $x + 3$ ...
6
votes
1answer
114 views

Permutations that preserve all algebraic relations between the roots of a polynomial

When trying to answer the question of whether a given equation can be solved with radicals, historically people have paid lots of attention to permutations that preserve all algebraic relations ...
12
votes
2answers
597 views

What is the mathematical intuition behind àl-jàbrà?

The term algebra comes from the arabic term àl-jàbrà that means "to force", "to restore". Over centuries mathematicians, in east and west, celebrate by this term mathematical disciplines. What is ...
2
votes
2answers
88 views

Why the terms “unit” and “irreducible”?

I'm trying to understand why in a ring we choose the names unit to an invertible element and irreducible element in this definition Maybe historical reasons? For example, I suppose the second ...
8
votes
4answers
437 views

Connections between number theory and abstract algebra.

I haven't taken abstract algebra yet, but I am curious about connections between number theory and abstract algebra. Do the proofs of things like Fermat's little theorem, the law of quadratic ...
3
votes
0answers
177 views

Which theorem did Poincaré prove?

Two related elementary facts in group theory are sometimes called Poincaré's theorems. If $H\lneq G$ and $[G:H]<\infty$, then there is $N\leq H$, $N\lhd G$ such that $[G:N]<\infty$. The ...
10
votes
4answers
430 views

Is the theory of dual numbers strong enough to develop real analysis, and does it resemble Newton's historical method for doing calculus?

I've been interested in non-standard analysis recently. I was reading up on it and noticed the following interesting comment on the Wikipedia page about hyperreal numbers, right after giving an ...
12
votes
1answer
147 views

Original Formulation of Hilbert's 14th Problem

I have a problem seeing how the original formulation of Hilbert's 14th Problem is "the same" as the one found on wikipedia. Hopefully someone in here can help me with that. Let me quote Hilbert first: ...
26
votes
3answers
743 views

Where does the word “torsion” in algebra come from?

Torsion is used to refer to elements of finite order under some binary operation. It doesn't seem to bear any relation to the ordinary everyday use of the word or with its use in differential geometry ...
0
votes
1answer
81 views

What type of famous equation except diophantine equation such that no algorithm can exist to determine whether there is a solution?

What type of famous equation except diophantine equation such that no algorithm can exist to determine whether there is a solution? I know that if these equation have a solution, then it could be ...
1
vote
0answers
114 views

Etymology of algebra (as k-algebra)?

Why algebra (over a field) is called "algebra?" (My random guess is that it's a back-formation of some algebras, chopping adjectives from say Lie algebra or Clifford algebra, etc.) And when was that ...
19
votes
1answer
339 views

When, and by whom, was “$\mathbb{C}$ is algebraically closed” dubbed the “fundamental theorem of algebra”?

Wikipedia has this enigmatic sentence on the page for the fundamental theorem of algebra: ...its name was given at a time when the study of algebra was mainly concerned with the solutions of ...
48
votes
1answer
1k views

How was the Monster's existence originally suspected?

I've read in many places that the Monster group was suspected to exist before it was actually proven to exist, and further that many of its properties were deduced contingent upon existence. For ...
11
votes
4answers
398 views

Why the terminology “monoid”?

As I am not a native English speaker, I sometimes am bothered a little with the word "monoid", which is by definition a semigroup with identity. But why this terminology? I searched some ...
1
vote
2answers
233 views

Could someone give me an example of an algebraic variety and explain what it is

I've read the wikipedia article but I don't know what an affine plane is and the definition/example did not seem clear. What I know is that in the 1880s mathematicians like Hilbert, Kronecker, Lasky ...
2
votes
1answer
159 views

What does Tarski mean by a “tautological operation” on a Boolean algebra?

I am reading Part II of Chin and Tarski's "Distributive and Modular Laws in the Arithmetic of Relation Algebras". In the beginning of section 4, the authors say "In general, if $\odot$ is a binary ...
7
votes
2answers
222 views

Terminology: center (of a group, of a ring, …)

What is the etymology of the word "center" as used in abstract algebra, e.g. the center of a group, or of an algebra? My best guess is that it might've come from matrix algebras, where often the ...
6
votes
1answer
281 views

Why are modules called modules?

I know that a module is a generalization of a vector space, but I would like to know why are modules called modules? Thanks for your kindly help.
6
votes
1answer
248 views

When did free modules first appear?

We study free modules in a Modern Algebra course or by reading a book on Algebra. In any case a free module looks like a vector space, for we consider the generating set and basis... My questions are ...
68
votes
1answer
6k views

Why are rings called rings?

I've done some search in Internet and other sources about this question. Why the name ring to this particular object? Just curiosity. Thanks.
11
votes
1answer
1k views

Proof of Euler's Theorem without abstract algebra?

Every proof I've seen of Euler's Theorem (that $\gcd(a,m) = 1 \implies a^{\phi(m)} \equiv 1 \pmod m$) involves the fact that the units of $\mathbb{Z}/m\mathbb{Z}$ form a group of order $\phi(m)$. ...
7
votes
3answers
667 views

Where is the name “coset” in group theory from?

One of the most important application of "coset", I think, is to prove the Lagrange's theorem, which was not originally stated in the group theoretic terms. In some textbooks I have read about ...
43
votes
6answers
2k views

What kind of “symmetry” is the symmetric group about?

There are two concepts which are very similar literally in abstract algebra: symmetric group and symmetry group. By definition, the symmetric group on a set is the group consisting of all bijections ...
8
votes
5answers
865 views

Historical textbook on group theory/algebra

Recently I have started reading about some of the history of mathematics in order to better understand things. A lot of ideas in algebra come from trying to understand the problem of finding ...
0
votes
1answer
153 views

Alternative, consistent frameworks of mathematics with isomorphic mappings to physical phenomenon

A friend of mine who is quite an aggressive Nominalist told me the other day: "Mathematics and numbers are arbitrary; they can accurately predict physical systems in real life only because they are ...
4
votes
2answers
851 views

Supplement for Jacobson's Basic Algebra I

I am going to be taking a first course in abstract algebra this fall and we are using Jacobson's Basic Algebra I for the course text. It looks like a good textbook, but it is lacking a lot of ...
3
votes
3answers
1k views

What is Modern Mathematics? Is this an exact concept with a clear meaning? [closed]

Motivated by this question I would like to know whether there is an exact definition of modern mathematics. In which point in time (century, decade) does one think, when speaking about modern ...