Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [abstract-algebra]

For questions about monoids, groups, rings, modules, fields, vector spaces, algebras over fields, various types of lattices, and other such algebraic objects. Associate with related tags like [group-theory], [ring-theory], [modules], etc. as necessary to clarify which topic of abstract algebra is most related to your question and help other users when searching.

Abstract algebra is the study of algebraic objects, i.e. sets endowed with one or more operations on the elements of those sets. In particular, the study of abstract algebra considers the algebraic structures and properties of which such operations induce. It can be considered as the generalization of the study of the algebraic structure of the integers and real numbers (arithmetic), or the study of matrices and vector spaces (linear algebra).

Some algebraic objects are monoids, groups, rings, fields, vector spaces, modules, algebras, and categories, among many other less prominent objects.

Examples

  1. The set of non-negative integers $\mathbb{N} = \{0,1,2,3,\dotsc\}$ is a monoid under the operation $+$.

  2. The integers $\mathbb{Z} = \{\dotsc,-1,0,1,\dotsc\}$ under the binary operation of $+$ form a group.

  3. Furthermore, $\mathbb{Z}$ has the structure of a ring when you consider it as being equipped with both addition and multiplication.

  4. The real numbers $\mathbb{R}$ with their usual addition and multiplication form a field.

  5. The set of $n\times n$ matrices with entries in $\mathbb{R}$ with matrix addition and multiplication form a ring.

  6. The set of $1\times n$ vectors over the real numbers, with vector addition, and multiplication by elements of the $n\times n$ real matrices on the right are an example of a module for the ring of matrices.

In addition to studying the objects themselves, abstract algebra considers homomorphisms between the objects and various constructions and tools, which are useful for studying the objects.

85022 questions
6
votes
3 answers

How to define indeterminates formally?

Indeterminate are used in algebra to define polynomials and formal power series. Here is the definition of polynomials: A polynomial in an indeterminate X is an expression of the form $ a_0 + a_1X + a_2X^2 + \ldots + a_nX^n,$ where the $a_i$ are…
newvie
  • 63
6
votes
1 answer

Determinant map is homomorphism and surjective.

I just came from a course of abstract algebra, and my teacher told us that the determinant map $\det : GL(n, \mathbb{R}) \to \mathbb{R}^\times$ is a surjective homomorphism. Here, $GL(n, \mathbb{R}) = $ the set of $(n \times n)$ matrices $M$ such…
user230283
6
votes
4 answers

Finite Generated Abelian Torsion-Free Group is a Free Abelian Group

I am trying to prove that every Finite Generated Abelian Torsion-Free Group is a Free Abelian Group. In order to do this, I am trying to show that if $\{x_1, \dots, x_n\}$ is a minimal generator of the group and if $n_1x_1 + \dots n_n x_n=0$ then…
user79594
6
votes
1 answer

the unit group of an infinite field cannot be cyclic

It is well-known that the unit group of a finite field is a finite cyclic group. But for infinite fields, e.g., $\mathbb{Q}$ or $\mathbb{R}$, the unit groups are not cyclic. I heard this fact in my class and I'm trying to prove it. Assuming the unit…
user258589
  • 61
6
votes
2 answers

Why minus times minus needs to be plus?

Possible Duplicate: Why negative times negative = positive? An Abstract Algebra text book has a sentence on its 1st chapter about natural numbers that i cannot get around easily. The sentence reads "Why minus multiplied by minus needs to be plus…
marcel
  • 69
6
votes
1 answer

How to find automorphism of a particular order.

Given a finite group if the automorphism group is known is it possible to write down all the automorphisms with respective orders? For example say the group $Z_{p^{2}}$ has automorphism group isomorphic to $Z_{p(p-1)}$ and I…
user118494
  • 5,837
6
votes
1 answer

Transcendental Extensions. $F(\alpha)$ isomorphic to $F(x)$

Let $E$ be an extension field of $F$ and $\alpha \in E$. Then $\alpha$ is transcendental over $F$ if and only if $F(\alpha)$ is isomorphic to $F(x)$, the field of fractions of $F[x]$. This was a theorem in an abstract algebra textbook with a very…
Cay
  • 395
6
votes
2 answers

If $f$ is a group homomorphism from $(\mathbb{Z},+)$ to $(\mathbb{Q}-\{0\},.)$ such that $f(2)=\frac{1}{3}$, then find $f(-8)$.

I came across the following question: If $f$ is a group homomorphism from $(\mathbb{Z},+)$ to $(\mathbb{Q}-\{0\},.)$ such that $f(2)=\frac{1}{3}$, then what is the value of $f(-8)$? By property of group homomorphism, we can write - $f(8) =…
Ritu
  • 1,663
6
votes
1 answer

On Weyl algebras and their centers over fields of positive characteristic.

I was trying to read some notes leading up the Dixmier conjecture, but I was hoping to see clarify a lemma. Suppose you have the Weyl algebra $A_n$ over a field $k$ of positive characteristic $p$. Why is the center of the Weyl algebra isomorphic to…
Tiffany Hwang
  • 1,199
6
votes
2 answers

Associativity: why does $((a \ast b) \ast c) = (a \ast (b \ast c))$ mean we can bracket longer expressions however we like?

Possible Duplicate: How does one actually show from associativity that one can drop parentheses? In some ways this question seems obvious, but it is rarely covered. The normal definition given for a binary operation $\ast$ on a set $S$ is that,…
John Gowers
  • 24,959
6
votes
1 answer

$x^p-x-1$ is irreducible over $\mathbb{Q}$[x]

For any prime p, prove that $x^p-x-1$ is irreducible over $\mathbb{Q}$[x]. (In a field of characteristic p this is true). I asummed exist root in $\mathbb{Q}$, let's call $\frac{\alpha}{\beta} \in \mathbb{Q}$. Then following that $\frac{\alpha…
Angelo
  • 189
6
votes
4 answers

Tensor product between quaternions and complex numbers.

Let $H$ be the Hamiltonian quaternions, $\mathbb C$ the complex numbers and $\mathbb R$ the real numbers. Identify $H\otimes_{\mathbb R} \mathbb C$ in familiar terms. This is an exercise of Modern Algebra of Garrett Birkhoff. I think it may have…
zacarias
  • 3,158
6
votes
3 answers

What about my proof is "nonsense"?

I am working on a question from Fraleigh's "A First Course In Abstract Algebra": A torsion group is a group all of whose elements have finite order. A group is torsion free if the identity element is the only element of finite order. A student…
Dair
  • 3,064
6
votes
3 answers

Orthogonal idempotents give direct sum?

Say $x$ and $x'$ are orthogonal idempotents so $x+x'=1$ and $xx'=0$ in some commutative ring $A$. Then for any $a\in A$, $a=ax+ax'$, so $A=Ax+Ax'$. Why does $Ax\cap Ax'=0$? If $y=ax=a'x'$, then $y^2=aa'xx'=0$. Maybe I'm missing something, but does…
miles
  • 63
6
votes
5 answers

$A/B \cong A \iff B = \{e\}$ for groups $A,B$?

Let $A$ and $B$ be groups (can be infinite) Is it true that $A/B$ isomorphic to $A$ $\Leftrightarrow$ $B=\{e\}$ I didn't find a way to prove it. Thanks
Ben
  • 71
1 2 3
…
99 100