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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.

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Is $a + b5^{1/4} ; a,b \in \Bbb{Z}$ an Integral domain? Closed under multiplication?

Problem 6d in Section 1.2 from "A Survey of Modern Algebra" - Birkhoff, Mac Lane Are the following sets of real numbers integral domains? (d) all real numbers $a + b5^{1/4} $ where a and b are integers. My question is more specifically about…
PossiblyDakota
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Prove that $ \ \mathbb{Q}(\zeta_n) /\mathbb{Q} \ $ is finite

(a) For $ \ n \in \mathbb{N} \ $ , let $ \ \zeta_n=e^{\large \frac{2 \pi \large i}{n}} \in \mathbb{C} \ $, Prove that $ \ \mathbb{Q}(\zeta_n) /\mathbb{Q} \ $ is finite. (b) Let $ \ \mathbb{Q}(\zeta_{\infty}) =\cup_{n \in \mathbb{N}}…
MAS
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Generator of $\mathbb Z/m\mathbb Z\times\mathbb Z/n\mathbb Z$ for coprime $n$ and $m$.

For coprime $n$ and $m$ we have that $\mathbb Z/m\mathbb Z\times\mathbb Z/n\mathbb Z\cong\mathbb Z/mn\mathbb Z$. An isomorphism is given by $$a+mn\mathbb Z\mapsto (a+m\mathbb Z,a+n\mathbb Z)$$ Question: Since the cyclic group is generated by (for…
Buh
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Which of the following statements is not true?

Suppose that $(\mathbb Q,+)$ be the additive rational group and $H$ a subgroup of it. Which of the following statements is not true? (a) If $\mathbb Q/H\cong \mathbb Q$, then $H=0$. (b) If $H\neq 0$, then every proper subgroup of $\mathbb Q/H$ is of…
Aliakbar
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What does "copies" of a Ring/Module etc. mean

Could someone explain me what the word "copies" in Terms of Rings/Vectorspaces/Modules means? E.g in the context "For a ring R, the smallest subring containing 1 is called the characteristic subring of R. It can be obtained by adding copies of 1 and…
MasterPI
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What is class equation of A5

I tried to derive it and I got 60 = 1+15+20+24. But we know that for every a in G; o(a)/o(G) and 24 does not divide 60 then how it is possible class equation. Your small hint might be helpful to me. Thanks.
Sonu Lamba
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If $\phi:G \to \overline G$ is group homomorphism, then prove that $|\phi(G)|$ divides $|G|$.

If $\phi:G \to \overline G$ is group homomorphism, then prove that $|\phi(G)|$ divides $|G|$, where $|G|$ is finite. One way to prove this is this: we know that $\phi:G \to \phi(G)$ is an $n$-to-$1$ mapping, where $n=\lvert\ker \phi\rvert$. So…
ramanujan
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Finding the multiplicity of a root in a finite field.

How many roots does $\mathbb{F_{16}}$ have in the polynomial $x^4-1$? What I've done: $x$ is root iff $x^4=1$ iff $x$ has order $4$,$2$ or $1$. However $\mathbb{F_{16}}-\{0\}=\langle \beta \rangle$ (because of the primitive order element) and…
mathie12
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$E/K$ is a field extension. Suppose $\exists m\in Z, \forall x\in E, [K(x):K]\leq m$. Then $E/K$ is a finite extension?

$E/K$ is a field extension. Suppose $\exists m\in Z_{>0}, \forall x\in E, [K(x):K]\leq m$. $\textbf{Q1:}$ Do I need separability to deduce that $E/K$ is a finite extension of degree at most $m$?(I did use separability to deduce simple extension and…
user45765
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Embedding of this free algebra

Let $V$ be a variety of algebra. Prove $F(X)$, the free algebra in $V$ with basis $X$ can be embedded in $F(Y)$ if $X \subset Y$. I found this question on Chegg here while trying to study, but the answer is absolutely incorrect. They literally…
user559412
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Taylor's theorem in field of characteristic $0$

If $ K $ is a field of characteristic $ 0 $ then for all polynomials $ f[X] $ over $ K $ and elements $ x, h \in K $, we have $$ f(x + h) = f(x) + f'(x) h + \frac{1}{2} f''(x)h^2 + \cdots $$ This doesn't seem so hard (but tedious) to prove using…
user228960
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Trace of outer power Endomorphism eqauls coefficients of characteristic Polynomial

For a Vector Space over the Complexe Numbers and a linear and diagonalizable map $g:V\rightarrow V$ the trace of ${ \wedge }^{ r }g:{ \wedge }^{ r }V\rightarrow { \wedge }^{ r }V$ is given by the Coefficient of the characteristic polynomial ${ x…
johnka
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For any integer $n \ge 3 $, prove that $D_n$ has a subgroup of order 4 if and only if n is even

Suppose $H \lt D_n$, $|H|=4$ Since $|D_n|=2n$, by Lagrange's Theorem $$4|2n$$ $$2n=4k $$ for some $\in \Bbb Z^+$ $$n=2k$$ thus $n$ is even Conversely suppose $n$ is even. then how to show $D_n$ has subgroup of order 4 ? please give me a hint…
fivestar
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R is semisimple if and only if every R module is projective

I am trying to show that $R$ is semisimple if and only if every R module is projective. What I know is that if $R$ is semisimple then every $R$ module $M$ is semisimple which implies $M$ can be expressed as a direct sum of simple modules i.e…
TheGeometer
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show that orbits have the same order under the normal subgroup of a transitive group

Let $G$ be a group that works transitively on $X$, and let $N$ be a normal subgroup of $G$. Show that the orbits of $X$ under $N$ have the same order, that is $\operatorname{ord}(Nx)=\operatorname{ord}(Ny)$ for all $x,y\in X$. I'm not sure how to…
Sha Vuklia
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