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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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If a polynomial does not have roots, does that imply it is irreducible?

One of my homework problems wants me explain why $x^2 +2$ is irreducible in $\mathbb{Z}_5$. The possible roots of $x^2+2$ are $x=\pm \sqrt{-2} \notin \mathbb{Z}_5$. Is it enough to say that since $x^2+2$ has no roots in $\mathbb{Z}_5$, it must be…
TheMobiusLoops
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Prove that a nonzero homomorphic image of a local ring is a local ring

The problem says that prove that a nonzero homomorphic image of a local ring is a local ring. Could you give me a scratch of a proof for this or maybe a full answer if you don't mind?
le4m
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Show that $F(E^p)$ consists of all linear combinations of elements in $E^p$ with coefficients in $F$.

Let $E$ be a finite extension of a field $F$ of prime characteristic $p$, and let $K=F(E^p)$ be the subfield of $E$ obtained from $F$ by adjoining the $p$th power of all elements of $E$. Show that $F(E^p)$ consists of all linear combinations of…
Idonknow
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Divisors of zero in $ \mathbb Z_{p^k}$

Let $p$ be a prime numer and let $k$ be a natural numer such that $k\geq 2$. I wish to descripe all zero's divisors in $\mathbb Z_{p^k}$. Obviously elements of the form $np$, where $n=0,...,p^{k-1}-1$, are zero divisors, because $p^k|np^k$. Are…
Alex
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Boolean ring is commutative in a different manner

Boolean ring $R$ is a ring where $x^2 =x$ $\forall$ $x \in R$. I want to show that a Boolean ring is commutative. I have done the following. $$x + x = ( x + x )^2 = x^2 + x^2 + x^2 + x^2 = x + x + x + x \Rightarrow 2x =0 $$ So any element of $R$ is…
Supriyo
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Subfields of an Infinite field of char p

Do all infinite fields of char p contain a subfield isomorphic to $F_{p}(x)$?
Mykie
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For $n$ at least 5, the index of a subgroup of $Alt(n)$ is at least $n$.

Can someone can help me get started with this problems? Prove that $A_n$ does not have a proper subgroup of index less than $n$ for all $n \geq 5$. I followed Robert's prove, but I was not able to sort out why it can't be $A_n$. Could someone help…
Tumbleweed
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Subgroups regarding group axioms

Practice Question Prove that a group with two elements of order 2 that commute must have a subgroup of order 4? I went with the approach that a group with order 2, can generally not exist under group axioms unless there is a subgroup with order 4.…
Aspiring Mathematician
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I is a prime ideal iff R-I is multiplicative

Suppose $R$ is a commutative ring with unity and that $I$ is an ideal of $R$. Then $I$ is a prime ideal iff $R-I$ is multiplicative (if $a,b\in R-I$, then $ab\in R-I$). So far I have been able to prove the forward direction. If $ab\in I$, then $a\in…
Tim
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Ideal subgroup of the additive group of a ring (R,+, *)

I just started learning ring theory. My professor defined an ideal as follows: An $\textbf{ideal}$ in a ring R is a nonempty subset $I\subseteq R$ such that if $a\in I$ and $r\in R$, then $ar,ra\in I$ and if $a,b\in I$, then $(a+b)\in I$. I want to…
Dan
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What is the Ambient field of rational functions in $n$ independent variables over $\mathbb{Q}(x_{n+1} , . . . , x_m)$?

I have been reading, and I am unsure what is meant by the following, Choose $m \geq n$ positive integers. Let $F$ be an ambient field of rational functions in $n$ independent variables over $\mathbb{Q}(x_{n+1} , . . . , x_m)$. Could someone please…
user93826
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necessary separability condition?

I wonder if and where the separability condition on $L$ is used in the following theorem from Lang's Algebraic Number Theory p. 7. I suspect it is necessary to see that a subring of the finite extension $L$ is a finitely generated $A$-module, but…
Peter Patzt
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If $R$ is a ring without divisors of $0$ (not necessarily commutative), is it true that for every $x\neq 0$, $xR=Rx$?

Let $R$ be a ring (not necessarily commutative), is it true that for every $x\neq 0$, $xR=Rx$?. What I've tried so far was: Let $y\in Rx$. Then $y=rx$ for every $r\neq 0,~r\in R$. Then $xrx\in xR$. $\therefore yr\in xR.$ I'm not sure, but couldn't…
Fabrizio Gambelín
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Prove Hom$_{\mathbb{Z}}(\mathbb{Z}/n \mathbb{Z},A) \cong A_n$

Let $A$ be a $\mathbb{Z}$-module, let $a$ be an element of $A$ and let $n$ be a positive integer. Prove that the map $\phi_a:\mathbb{Z}/n \mathbb{Z} \rightarrow A$ given by $\phi(\bar{k})=ka$ is a well-defined $\mathbb{Z}$-module homomorphism iff…
Idonknow
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Distributive Property - A Converse Theorem

Suppose that $\ast$ distributes over $+$, where $0$ is the additive identity. We can conclude the following. $a \ast b = a \ast \left( b + 0\right)$ $a \ast b = a \ast b + a \ast 0$ $\therefore a \ast 0 = 0$ In other words, the additive identity…
Ryan
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