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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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Every element of $Z (\sqrt{-5})$ is factorable into irreducible factors.

I am trying to proof that each element of $R :=\mathbb{Z} (\sqrt{-5})$ is factorable into irreducible elements. Let $x \in R, x = AB$ with $A = (a + b \sqrt{-5} ),$ $B = (c + d \sqrt{-5})$. If $A = \pm 1$ or $B = \pm 1$ then $x$ irreducible and the…
newbie
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How to find an element in a field with norm of a specific number?

Finding the norm of an element in a field, say, $\mathbb{Z}[\sqrt{19}]$ is rather easy, it just requires the computation of $(a+b\sqrt{19})(a-b\sqrt{19})$. However, given a value, say, 2, how do we find elements in $\mathbb{Z}[\sqrt{19}]$ that have…
DurhamManiac
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$S_n$ acts faithfully on a finite set $X$ implies an orbit of size $n$

If $S_n$ acts faithfully on a finite set $X$, must there be an orbit of size $n$? The context of the question is the following: suppose a Galois extension $E/F$ has Galois group $G$ isomorphic to the symmetric group $S_n$. Let $f \in F[x]$ be a…
MT_
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If G is an abelian group and $n \in \Bbb N$, show that $\phi :G \rightarrow G$ defined by $ g \mapsto g^n$ is a group homomorphism

Quoting "If G is an abelian group and $n \in \Bbb N$, show that $\phi :G \rightarrow G$ defined by $ g \mapsto g^n$ is a group homomorphism." My take here is that We cannot establish directly that $\phi (a \circ b) = (a \circ b)^n = a^n . b^n =…
gegu
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Determining the number of functions in Galois Group

I am supposed to determine the number of functions in Gal $(\sqrt[4]{3},i)$ without solving for what the exact functions are. That is a separate problems. Below is the general outline of my proof. I attempt to follow an example in the text and am…
NUG
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Cosets of S3 and Permutations

Quoting: "Let H be the subgroup of S3 defined by the permutations {(1); (123); (132)}. The left cosets of H are (1)H = (123)H = (132)H = {(1); (123); (132)} (12)H = (13)H = (23)H = {(12); (13); (23)} " I am a bit stock here, I am not understanding…
gegu
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some question related to $\mathbb{Z}[\sqrt{-5}]$ and UFD

could any one give me examples/proofs/counter examples against or for of the followings? 1.Homomorphic image of a UFD is again a UFD 2.The element $2\in\mathbb{Z}[\sqrt{-5}]$ is irreducible 3.Units of the ring $\mathbb{Z}[\sqrt{-5}]$ are units of…
Myshkin
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If $S$ is a ring with the property that $s=s^2$ for each $s \in S$, then is $S$ commutative?

If $S$ is a ring with the property that $s=s^2$ for each $s \in S$, then is $S$ commutative? I know $st=(st)^2=stst$. On the other hand, $st=s^2t^2$. Hence, $$stst=sstt$$ If $S$ is a division ring, then by cancellation we can have $st=ts$. But how…
SHBaoS
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How many additive abelian groups $G$ of order 16 have the property such $x+x+x+x=0$

Up to isomorphic, how many additive abelian groups $G$ of order 16 have the property such that $x+x+x+x=0$, for each $x$ in G? My question is that which theorem can be used? My answer is 3. Is that right?
Jill Clover
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Describe the cosets of the subgroup ℤ of ℝ

Please note that this is a problem from a Book of Abstract Algebra by Pinter. Now, in his book, Pinter refers to ℤ as the additive group of integers, which he alternatively denotes as <ℤ,+>. Also, Pinter refers to ℝ as the additive group of real…
Kernel Sohcahtoa
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decompose a group into semidirect product

Question: $$G=\left\{\begin{bmatrix} 1&a&b \\0&1&c\\0&0&1\end{bmatrix},a,b,c\in Z_{p}\right\}$$ Prove that $G=(Z_{p}\times Z_{p})\rtimes Z_{p}$. So for our purpose,we must find a normal subgroup of $G$ which is isomorphic to $Z_{p}\times…
Jack
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How should I write down the alternating group $A_3$?

I didn't understand how to write down the alternating group A3. Is this the group consisting of only the even permutations? Also, what familiar group is this isomorphic to?
Anastasia
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Let $\alpha$ be a root of $X^3+X^2-2X+1\in\mathbb{Q}[X]$.

Here is a question in my homework. Let $\alpha$ be a root of $X^3+X^2-2X+1\in\mathbb{Q}[X]$. Express $(1-\alpha^2)^{-1}$ as a $\mathbb{Q}$-linear combination of $1$, $\alpha$ and $\alpha^2$. Justify the assertion that the cubic is irreducible…
Chuwei Zhang
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Infinite cyclic group isomorphism.

I have one probably basic question, but still it bothers me how to show it. Namely, if two groups are isomorphic (i.e. there is a bijective group homomorphism between them) and if one of them is infinite cyclic (precisely, in my case it is discussed…
edward_scissorhands
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A Duality between Hypercomplex Algebras

Consider the commutative, unital algebras $\mathbb{R}(i), \mathbb{R}(\epsilon)$ and $\mathbb{R}(\eta)$, where the adjunctions satisfy $i^{2} = -1, \epsilon^{2} = 0$ and $\eta^{2} = 1$ (but $i, \epsilon$ and $\eta$ are not elements of $\mathbb{R}$).…
user02138
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