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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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Can one construct such a morphism of fields?

Let $K$ be the field $\mathbb{Q}(\sqrt[3]{2})$. I want to construct an explicit morphism from $K$ to the fraction field of $$\mathbb{Q}[X,Y,Z]/(X^3 + 2Y^3 + 4Z^3 - 6XYZ)$$ but this doesn't seem to be that easy. Can someone help me? Of course I just…
Evariste
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How is this a good proof

To me this just reads: We suppose this and that and because we supposed it, it must be true. Why does this show that $H$ is closed under inverse? Why does it show that $H$ is closed under the operation? It doesn't, it just assumes it. Am I…
user3200098
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A group-like structure where the existence of the inverse element is replaced by divisibility

Say that a Positive Group is a poset $(G,\geq)$ with an operator $\cdot$ s.t. (Closure) For all $a, b \in G$ we have $a \cdot b \in G$ (Associativity) For all $a, b, c \in G$ we have $a \cdot (b \cdot c) = (a \cdot b) \cdot c$ (Identity element)…
exk
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Condition under which the subring of multiples of an element equal the ring itself?

Let $R$ be a commutative ring with identity with $b\in R$. Let $T$ be the subring of all multiples of $b$, $T=\{r\cdot b : r \in R\}$. If $u$ is a unit in $R$ with $u \in T$, prove that $T=R$. Could you help me some suggestions? I really have no…
ZHJ
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Suppose G has order 4, but contains no element of order 4. A) prove that no element of G has order 3.?

Also, B) Explain why every non-identity element of G has order 2. C) Denote the elements of G by e, a, b, c and write out the operation table for G.
Jewelss
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Relatively prime numbers and their gcd

I have a proof question that I'm struggling on for my abstract algebra class. Suppose that $a, b, c, d$ are integers with $a\neq 0$. Suppose that $a$ and $b$ are relatively prime. Part (i) If $d|a$ then, $b$ & $d$ are relatively prime. I've…
George1811
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For a polynomial ring $A[x]$, is $p(x) = x$ always irreducible?

I am a little confused by the definition of irreducible elements. In particular, for a polynomial ring $A[x]$, is $p(x) = x$ always irreducible? Or does it depend on the properties of $A$?
Jean Valjean
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A question about isomorphism

Let $p$ be a prime and let $\Bbb Z(p)$ be the ring of integers mod $p$. Define a function $f:\Bbb Z(p)\to \Bbb Z(p)$ by $$f(a)=\begin{cases}0, &a=0\\ a^{-1},&a \ne0\end{cases}$$ Find all primes $p$ for which $f$ is a ring isomorphism. I have…
user121819
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Where can I make abstract algebra tangible?

In my text-book there is just an endless stream of theorems and it's just a big hot bowl of algebra with literally nothing to compare it all to. I know that a relation is a specific form of connection between sets but my book decided that explaining…
Paze
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Prove about cyclic

Prove that every finite subgroup of the multiplicative group $T=\{z \in \Bbb C||z|=1\}$ is cyclic. I was thinking to prove that the order of every subgroup of $T$ is prime, then they are all cyclic. But I can't. Could somebody give me some hints.…
user121819
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Determining If Something Is A Set Is A Ring.

I had a question that I don't understand. I was looking through my abstract algebra textbook and I did a couple problem and I got most of the the problems right except for this one: $$R = {(a + b\sqrt[3]{3}}) :a,b\in\mathbb{Q}$$ It might be a dumb…
chris
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Prove that if $d$ divides $n$ then $φ(d)$ divide $φ(n)$ for $φ$ denotes Euler’s φ-function.

Prove that if $d$ divides $n$ then $φ(d)$ divide $φ(n)$ for $φ$ denotes Euler’s $φ$-function. I know that $d|n$ mean there exists some integer $k$ such that $n=kd$, but how can I use this to prove $φ(d)$ divide $φ(n)$
Diane Vanderwaif
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Goursat's Lemma

Referring to p. 75 in Lang's Algebra, the statement of Goursat's lemma seems unclear to me. What exactly are the projections $p_1$ and $p_2$? Are they the standard projections $(x_1,x_2) \stackrel{p_i}{\mapsto} x_i$, for $i=1, 2$? If yes, then $p_i$…
Manos
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Converting From One Representation of a Field Element to Another

Let $A \subseteq B \subseteq C$ be fields and let $\alpha$, $\beta$, $\gamma$ be such that $A(\alpha) = B$, $B(\beta) = C$, $A(\gamma) = C$. Assume $B$ and $C$ have finite degree over $A$. Let $m(\alpha,A)$ be the minimal polynomial of $\alpha$…
maxpower
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Let $G$ be a finite abelian group of odd order. Which of the following define an automorphism of $G$?

Let $G$ be a finite abelian group of odd order. Which of the following define an automorphism of $G$? a. The map $x→ x^{−1}$ for all x ∈ G. b. The map $x→ x^2$ for all x ∈ G. c. The map $x→ x^{−2}$ for all x ∈ G. I have verified that all of…
gubloo
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