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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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Determine if the Gaussian Integers $\mathbb{Z}[i]$ and $\mathbb{Z}\times\mathbb{Z}$ are isomorphic rings

Yet again I am having difficulty with another exercise from my abstract algebra class. It is a homework question that my professor came up with himself and is as follows, word by word: Let $R$ and $T$ be rings. A function $f:R\to T$ is a ring…
erik7970
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Show that a group homomorphism $f$ is the identity.

Suppose that $f$ is a group homomorphism from $\mathbb Z_7\times\mathbb Z_7$ to itself satisfying $f^5 = \operatorname{id}$ (where $f^5=f\circ f\circ f\circ f\circ f$). Show that $f$ is the identity.
Yeyeye
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$f: G→G$ defined by $f(x) =x^2$ is a homomorphism if and only if $G$ is abelian.

The function $f: G→G$ defined by $f(x) =x^2$ is a homomorphism if and only if $G$ is abelian. Can anyone give me any tips how to work on this question?
megan
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Prove that the index of the set of left coset in a cyclic group is finite

If G is cyclic , show |G/H| < ∞ for any subgroup H except the identity. I already know that any subgroup of a cyclic group is also cyclic but i have no idea how to prove a the quotient of G is finite especially when Lagrange theorem also only apply…
Minh Vu
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Why do we exclude $0$ and units from the definition of irreducible elements?

Why do we exclude $0$ and units from the definition of irreducible elements? More clearly how can it be shown that for the units and $0,$ $c=ab$ doesn't always imply either $a$ or $b$ is a unit (where $c = 0$ or $c$ is a unit)? For $0$ my guess is…
Sriti
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Two problems about Structure Theorem for finitely generated modules over PIDs

1) Let $A$ be a real 4 by 4 matrix. Supose $i,-i$ are the eigenvalues of $A$. Show that there exists an invertible matrix $P$ such that $PAP^{-1}$ is either $$\begin{pmatrix} 0&-1&0&0\\ 1&0&0&0\\ 0&1&0&-1\\ 0&0&1&0 \end{pmatrix},…
Ishigami
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Applications of the concept of homomorphism

What are some interesting applications of the concept of homomorphism? Example: If there is a homorphism from a ring $R$ to a ring $r$ then a solution to a polynomial equation in $R$ gives rise to a solution in $r$. e.g. if $f:R \rightarrow r$ and…
user1613
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Proving that $\mathbb{Z}[x]/\langle x^2 + 1\rangle$ is isomorphic to $\mathbb{Z}[i]$

I have some general idea of going about this. I'm trying to come up with a suitable mapping. I've come up with $\phi\colon a + \langle x^2 + 1\rangle\longmapsto a$ where $a$ is an irreducible element in $\mathbb{Z}[i]$ but this mapping has some…
Person
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How to prove these two groups are isomorphic?

Let $U_p$ be the subgroup of $\mathbb{C}^*$ satisfying that: for any $x\in U_p$, there exists an integer $n$ such that $x^{p^n}=1$. Let $K$ be a algebraic closed field with characteristic $0$. Denote $\mu_{p}$ be the subset of $K$ satisfying: for…
hxhxhx88
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Compute the splitting field

I have to compute the splitting field of $x^6-1 \in Q[x]$. I know that $x^6-1=(x+1)(x-1)(x^2-x+1)(x^2+x+1)$ but I don't know what to do after that. Please help me.
Breezy
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Show that $ord(ab) | \frac{mn}{\gcd(m,n)}$ and $\frac{mn}{\gcd(m,n)^2}|ord(ab)$.

Suppose $G$ is an abelian group. Define $ord(a)=m$ and $ord(b)=n$ where $a,b \in G$. Show that $ord(ab) | \dfrac{mn}{\gcd(m,n)}$ and $\dfrac{mn}{\gcd(m,n)^2}|ord(ab)$. Since $(ab)^{\dfrac{mn}{\gcd(m,n)}}=((a^m)^n(b^n)^m)^{\dfrac{1}{\gcd(m,n)}}=1$,…
Idonknow
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Exhibit the correspondence in the Correspondence Theorem.

Here is the whole problem. I answered the first two parts, but I can't get down the third part. Problem Consider the group $D_{4} = \langle x,y:x^{2}=1, y^{4}=1, yx=xy^{3}\rangle$ and the homomorphism $\Phi : D_{4} \rightarrow Aut(D_{4})$ defined…
NasuSama
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subfields of a finite subfield

I'm trying to work out all the subfield of a finite field with $3^7$ elements. So I've said that since every subfield of that field if of the form $3^n$ where $n$ is a positive divisor of $7$. so I got the finite subfields as the finite field with…
lard
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Prove that $H=0$ or $H=\mathbb{Q}$

Let $H$ be a subgroup of the additive group of rational numbers with the property that $\dfrac{1}{x} \in H$ for every non-zero element $x$ of $H$. Prove that $H=0$ or $H=\mathbb{Q}$ Let $x \in \mathbb{Q}$. Then $x=\dfrac{a}{b},a,b\in \mathbb{Z},b…
Idonknow
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Subrings of integral domains have the same identity element.

If $R$ is an integral domain and a subring $S$ has identity $1_S$, how would you show that $1_S=1_R$ (here $1_R$ is the identity of the ring $R$)? I am unsure about what an integral domain really is and how the subring comes into play here.
user72195
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