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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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Linear basis for a quotient ring

Question: Let $k$ be a real number, and let $A$ denote the ring $\mathbb{R}[x]/(x^2+k)$. Find an $\mathbb{R}$-linear basis for $A$ and describe the multiplication law in terms of this basis. I am not quite sure about the meaning of…
LaTeXFan
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Please help clear up some confusion I have about cosets

first my definition of a coset that I got is: Definition. Let $H$ be a subgroup of a group $G$. A left coset of $H$ in $G$ is a subset of $G$ that is of the form $xH$, where $x\in G$ and $$xH = \{y\in G | y = xh \text{ for some }h\in H\}.$$ And…
Deven Ware
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If $G_1$ and $G_2$ are simple groups, what can we say about normal subgroups of $G_1 \times G_2$?

If $G_1$ and $G_2$ are simple groups, what can we say about normal subgroups of $G_1 \times G_2$? I remember when I was taking Algebra I this was brought up in the class but at the time the professor left it for us to think over it. Well, I remember…
user66733
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Use Corollary 2 of lagrange's theorem to prove that the order u(n) is even when n>2

Use Corollary 2 of lagrange's theorem to prove that the order U(n) is even when n>2. Corollary 2: In a finite group, the order of each element of the group divides the order of the group. Group U(n) is operation muiltiplication mod n. And,…
Deep Blue
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Automorphism proof

Let $G$ be a finite group, $T$ an automorphism of $G$ with the property that $T(x) = x$ for $x \in G$ if and only if $x = e.$ Prove that every $g \in G$ can be represented as $g = x^{-1}T(x)$ for some $x \in G$. I am having trouble understanding…
user104235
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Extending algebra into field

$\newcommand\mee{\mathbin{\text{::}}}\newcommand\moo{\mathbin{\text{#}}}$ Let $\mathcal U$ be the collection of all finite subsets of $\mathbb N$. Let $\mee$ be a binary operation defined as: $$\begin{matrix} \mee:&\mathcal U\times\mathcal…
Carlos Eugenio Thompson Pinzón
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Prove that $b^2=a^2$

Let $G$ be a group of order $8$. Assume that there exists $a \in G$ such that $\lvert a\rvert =4$ and that no elements of $G$ has order $8$. Assume $\langle a \rangle \lhd G$, $b \notin \langle a\rangle$ and $b^2 \in \langle a\rangle$. Suppose…
Nadia C
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Uncertainty in the definition of "Dual Group"?

Referring to Lang's Algebra p. 46, given an abelian group $G$ of exponent $m$, the dual group is defined to be $Hom(G,Z_m)$ and is denoted by $G^{\wedge}$. This is were it does not feel right: every multiple of $m$ is an exponent as well and so we…
Manos
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If $G \cong H/K$, does it follow that $H \cong G \times K$?

It is very tempting to perform this step, but I feel like it is not true. I couldn't come up with a counterexample though.
user103461
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Existence of extension field containing all roots of a polynomial $f(x) \in \mathbb{F}[x]$.

This is not for homework, but I would just like a hint please. The question asks If $f(x) \in \mathbb{F}[x]$ has degree $n$, show that there exists an extension field $\mathbb{E}$ of $\mathbb{F}$ such that $\mathbb{E}$ contains all the roots of…
tylerc0816
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Showing that $\mathbb{Z}_3\times \mathbb{Z}_4$ is Isomorphic to $\mathbb{Z}_{12}$

How can I show that $\mathbb{Z}_3\times\mathbb{Z}_4$ is isomorphic to $\mathbb{Z}_{12}$ I found an order twelve generator, and that was the hint, but can I show it is an isomorphism without showing it's a bijective homomorphism, or is that the only…
user82004
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Group covered by finitely many cosets

This question appears in my textbook's exercises, who can help me prove it? If a group $G$ is the set-theoretic union of finitely-many cosets, $$G=x_1S_1\cup\cdots\cup x_nS_n$$ prove that at least one of the subgroups $S_i$ has finite index in…
python3
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Symmetric Difference Identity

Let $A + B = (A - B) \cup (B - A)$ also known as the symmetric difference. Look for the identity and let $e$ be the identity element $A + e = A$ $(A - e) \cup (e - A) = A$ Now there are two cases: $(A - e) = A$ This equation can be interpreted as…
Student
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Number of $Z$ homomorphisms from $Z_{n}$ to $Z_{m}$ using tensor product

It is a standard result that $\textrm{Hom}_{\mathbb{Z}}(\mathbb{Z}_{n},\mathbb{Z}_{m})=\mathbb{Z}_{(n,m)}$. Is it possible to show this using the following result: $\mathbb{Z}_{n} \otimes_{\mathbb{Z}} \mathbb{Z}_{m} = \mathbb{Z}_{(n,m)}$
user6495
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How is $e^{i2\pi /7}$ a solution to a quadratic over $\mathbb{Q}(\cos (2\pi/7))$?

I'm doing some exercises in the cyclotomic extension $\mathbb{Q}_7$. I have $\omega=e^{i2\pi /7}$, and I know $\omega+\omega^6=\omega+\omega^{-1}=2\cos(2\pi/7)$. My book says that then $\omega$ is a solution to a quadratic over $\mathbb{Q}(\cos 2\pi…
Eli
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