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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
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Any subfield of $\mathbb{C}$ must contain every rational number

I tried to prove that any subfield of $\mathbb{C}$ must contain every rational number by contradiction. Proof: Let $\mathbb{F}$ be any subfield of $\mathbb{C}$. Thus, $\mathbb{F}$ is itself a field under the usual operations of addition …
Ritu
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Are there cyclic, free modules where the generating element isn't a basis?

Let $R$ be a ring, and $M$ a nontrivial cyclic, free $R$-module. Let $m$ generate $M$, so that $M = Rm$. Is it then the case that $m$ forms a basis for $M$, so that $\mbox{ann}_{R}(m) = (0)$? I know that if $R$ is a domain or a commutative ring, it…
Elchanan Solomon
  • 30,807
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Assume that $(G, \ast)$ is a group and that every element $a \in G$ satisfies $a \ast a = 1$. Show that $(G, \ast)$ is abelian.

Assume that $(G, \ast)$ is a group and that every element $a \in G$ satisfies $a \ast a = 1$. Show that $(G, \ast)$ is abelian. To prove $(G,\ast)$ is abelian, we must show that it is commutative. Let $a,b \in G$. Then $a \ast b \in G$ and $(a…
St Vincent
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A field $F$ has at most a finite number of elements of order $\leq$ $n$ for any $n$ in integers.

A field $F$ has at most a finite number of elements of order $\leq n$ for any $n$ in integers. How can I prove this? I thought it's related to the fact that a polynomial of degree $n$ would have at most $n$ roots over $F$. But I am not sure. Would…
Heudsf
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How to determine injectivity and surjectivity for a map $\mathbb{Z}_n \to \mathbb{Z}_{n}$?

I know how to determine injectivity and surjectivity for maps between regular sets, but in this case I've got some problems. How can I solve this? Given the following map $\psi:\overline{x} \in \mathbb{Z}_{16}\mapsto…
BAD_SEED
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Show a rational function is transcendental over a field.

Let $u=\frac{x^3}{x+1}\in F(x)$, where $F(x)$ is the field of quotients of $F[x]$ ($F$ some field, $x$ an indeterminate over it). Show that $u$ is transcendental over $F$. This is an exercise in Hungerford. I'm having some trouble even grasping…
FPP
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Proper definition of a field?

I just read a very short definition of a field where it was said that a field is a set of elements $K$ with two maps from the field into the field itself, such that $K$ is an abelian group with $+$. $K \backslash \{0\}$ is an abelian group with…
user159356
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Solution of a polynomial of degree n with soluble galois group.

Background: Given the fundamental theorem of algebra every polynomial of degree n has n roots. From Galois Theory we know that we can only find exact solutions of polynomials if their corresponding Galois group is soluble. I am studying Galois…
nilo de roock
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How do I show $[\mathbb{Q}(\sqrt{2},\sqrt{3}):\mathbb{Q}]\geq[\mathbb{Q}(\sqrt2):\mathbb{Q}][\mathbb{Q}(\sqrt3):\mathbb{Q}]$?

I know how to show $[\mathbb{Q}(\sqrt{2},\sqrt{3}):\mathbb{Q}]\leq[\mathbb{Q}(\sqrt2):\mathbb{Q}][\mathbb{Q}(\sqrt3):\mathbb{Q}]$, but don't know how to show the converse inequality. $[\mathbb{Q}(\sqrt2):\mathbb{Q}]$ and…
user42383
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First Isomorphism Theorem to identify a quotient

I understand the notion of quotient groups quite well (I think), but I'm struggling a little bit with the following problem: Let $G$ denote the group of 2x2 invertible real upper triangular matrices, and $H\vartriangleleft G$ the subgroup with…
theage
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Every finite dimensional representation of an algebra has an irreducible sub representation

Let $V$ be a nonzero finite dimensional representation, i.e we have a homomorphism $\rho\colon A\rightarrow \text{End}_k(V)$, of an algebra $A$. I have to show that there is an irreducible sub representation. This is how wanted to do that: Let $v\in…
Badshah
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Solve simple xor equation

I have to solve few simple equation like this: $$ 117 = 86 \oplus x $$ I don't know how to move x to one side and calculate its value. How to do it?
andrzej1_1
  • 163
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Prove this matrix has infinite order

Consider the matrix: $M = \begin{pmatrix} 8&-3\\ 3&-1 \end{pmatrix}$ What are ways of showing $M$ has infinite order? I guess one is to find a closed form for the powers of $M$, but is there a more elegant way?
Learner
  • 227
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Homogeneous polynomial in $n$ variables of degree greater than $n(n-1)$ is in the ideal generated by the elementary symmetric polynomials

Let $f(x_1,...,x_n) \in \mathbb{Z}[x_1,...,x_n]$ be homogeneous of degree $d>n(n-1)$, i.e. $f$ is the sum of monomials of degree $d$. I am looking for a hint to prove that $f$ is in the ideal generated by the elementary symmetric polynomials…
Manos
  • 25,833
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Solving the Direct (or Inductive) Limit

Suppose we wish to compute the direct limit of the following. $$G\,\,\overset{M}{\longrightarrow}\,\, G\,\,\overset{M}{\longrightarrow}\,\, G\,\,\overset{M}{\longrightarrow}\,\, \cdots,$$ where each $G=\mathbb{Z}^d$, $d\in\mathbb{N}$, and the…
Eric
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