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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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What is the minimal polynomial of $\sqrt{3} + \sqrt[3] {2}$ over $\mathbb{Q}$?

What is the minimal polynomial of $\sqrt{3} + \sqrt[3]{2}$ over $\mathbb{Q}$? I know the basic idea of what a minimal polynomial is--it is the lowest degree monic polynomial in $\mathbb{Q}[x]$ that has the above as a root. But how do you go about…
setholopolus
  • 959
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For a Simple Group G, Z(Aut(G)) Is Trivial if and only if G is Non-Abelian.

Let $G$ be a simple group of order greater than $2$. Then $Z(Aut(G))$ is trivial if and only if $G$ is not Abelian. Let $H = Z(G)$ and let $g\in G, h\in H$. Then $gh = hg$ for every $g\in G$. So $ghg^{-1} = h$ for every $g\in G$ and we have…
Tanner Strunk
  • 985
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Proof only one finite field with p elements

I am not sure if it is possible with my experience to prove something like this but I would be interested in how such a proof would work. I am an undergraduate and have had a Modern Algebra course so my understanding may be limited. Essentially, I…
user43138
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prove that $Inn(A) \ne Aut(A)$ , A is abelian group

Let $A \ne \{e\}$ be abelian group which is not isomorphic to $Z_2 $. Prove that $Inn(A) \ne Aut(A) $. first, I have proved that $Inn(A) = \{id_a\} $. So, we need to prove that A got a not trivial automorphism. I consider 2 cases: 1) there exist…
user335501
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3 answers

Showing a congruence

How do we show that $(a_1+...+a_n)^2=(a_1^2+...+a_n^2)\mod 2$? I can see that this works when there are two terms, but I have trouble visualizing the left hand expansion with infinite terms. I guess this could call for a proof by induction, but once…
still_learning
  • 1,912
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Quaternions and Klein four group rings

I'm trying to prove that $\mathbb{H_{\mathbb{F}_2}}$, the ring of quaternions over the finite field $\mathbb{F}_2$, is isomorphic to the group ring $\mathbb{F}_2[V_4]$, where $V_4$ is the Klein-four group. As $\mathbb{H_{\mathbb{F}_2}} \cong…
cloudchamber
  • 565
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How many group homomorphisms are there from $\Bbb Z/2\Bbb Z \times \Bbb Z/2\Bbb Z$ to $S_{3}$

There is a similar problem that $\Bbb Z/2\Bbb Z \times \Bbb Z/2\Bbb Z$ to $S_{4}$. But I still confuse, This is what I tried: Let $f$ be homomorphism from $\Bbb Z/2\Bbb Z \times \Bbb Z/2\Bbb Z \to S_{3}$ then $\Bbb Z_2\times \Bbb Z_2/\ker f \cong…
Wade
  • 105
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Subgroups of $\mathbb Z \times(\mathbb Z/n\mathbb Z)$

So I am dealing with a problem from Dummit (specifically 2.1.7) and am having some issues. The part of the problem in question is: Prove the set of elements of the direct product $\mathbb{Z} \times (\mathbb{Z} / n \mathbb{Z})$ of infinite order…
KF Gauss
  • 207
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When the order of group is $p^3$, is this map a homomorphism?

Map is given, for any element $x \mapsto x^p$. I proved that the image of this map is contained in center of $G$. It is of course true when $G$ is commutative. So assume that is is not that case, $G/C(G)$ must be isomorphic to $Z/pZ\times Z/pZ$. Let…
Kim boeyri
  • 31
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If $G$ has order $p^2$, and $H$ is a subgroup of order $p$, then $H$ is normal (Searching for cleaner proof)

I'm self studying and came across the following problem that I'm having some trouble with. If $G$ has order $p^2$, and $H$ is a subgroup of order $p$, then $H$ is normal in $G$. I understand that this post shows a similar result. However, this is…
Nitin
  • 2,948
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problem on set of all matrices whose trace is zero

Let $T(n;\mathbb{R}) \subset M(n;\mathbb{R})$ denote the set of all matrices whose trace is zero. Write down a basis for $T(2;\mathbb{R})$. What is the quotient space $M(n;\mathbb{R})/T(n;\mathbb{R})$ isomorphic to? Answer of the first part is…
mintu
  • 91
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It is possible to define homomorphisms between vector spaces with different fields?

I understand that an homomorphism between vector spaces must preserve the sum between vectors and the scalar multiplication. By example, let vector spaces $(E,\Bbb Q)$ and $(F,\Bbb Q[\sqrt 2])$, then we can define something like $$f:E\to Q$$ such…
Masacroso
  • 30,417
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Filtered colimits or direct limits in DGA

Let $DGA(R)$ be the category of differential graded algebras (with unit) over a commutative ring $R$. I often read that this category obviously has all filtered colimits. Can someone explain them? Or can somebody explain direct limits in this…
tzpl11
  • 31
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The existence of an intermediate purely transcendental field extension.

Given a finite field extension $K\subset L$, prove there exists and intermediate field $M$ that is purely transcendental over $K$ (or $K=M$) and $[L:K]<\infty$. I feel like in concept the answer to this is fairly straightforward. After all, since…
rwmak
  • 167
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Find a subring of $\Bbb Z \oplus \Bbb Z$ that is not an ideal of $\Bbb Z \oplus \Bbb Z$.

Find a subring of $\Bbb Z \oplus \Bbb Z$ that is not an ideal of $\Bbb Z \oplus \Bbb Z$. I can't see any way a subring of $\Bbb Z \oplus \Bbb Z$ can NOT be an ideal. Subrings of $\Bbb Z \oplus \Bbb Z$ are of the form $n\Bbb Z \oplus k\Bbb Z$ where…
Oliver G
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