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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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Confused about the prime elements of a ring.

I am having a difficult time verifying if I am correct. I am working on a this released exam. This problem: Looking at this problem: Prime elements of $\mathbb{Q}$. I can rule out that any power of $2$ is a prime element of this ring as it is a…
Dair
  • 3,064
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Let $G$, $H$ be groups. Prove: $e_G \times H$ is normal in $G \times H$

I've proved that $e_G \times H$ is a subgroup of $G\times H$ and I know the various ways of defining normality. I am just having a bit of trouble writing this last part of my proof - any hints, tips etc. for showing that $e_G \times H$ is normal in…
user239484
  • 33
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List all the elements of ${\mathbb F_2}[X]/(X^3+X^2+1)$.

I just don't get this. Looking through past papers, I came across this problem, Q. Let $F_2$ be a field with 2 elements. Let $P=x^3+x^2+1\in F_2[X]$. $I$ is the ideal of $F_2[X]$ generated by $P$. List all elements of the factor ring $F_2[X]/I$. My…
Melba1993
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Abelianised dihedral group isomorphism

I have the dihedral group $$D_n =\langle x,y\mid x^n, y^2,(xy)^2\rangle$$ I need to show that the $D_n$ abelianised is isomorphic to $Z_2$ if $n$ is odd and $Z_2 \oplus Z_2$ if $n$ is even. How do I show this?
Al jabra
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Be $K$ of characteristic zero and $f(x) \in K[x]$. Prove that $f'(x) = 0$ then $f(x)$ is a constant polynomial.

Be $K$ of characteristic zero and $f(x) \in K[x]$. Prove that $f'(x) = 0$ then $f(x)$ is a constant polynomial. I know that the field of zero characteristic is a field where any sum of multiplicative identity element with itself, $1 + 1 + ... + 1$…
Danilo Santana
  • 165
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Extend a homomorphism

Assume $B,H,G$ are abelian groups, $f:B\rightarrow H$ is a surjective homomorphism, $H$ is a subgroup of $G$. My question is :is there an abelian group $A$ and a surjective homomorphism $g:A\rightarrow G$ satisfies: (1) $B$ is a subgroup of $A$. (2)…
Li Cong-xiao
  • 43
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Is the condition of PID necessary?

In Gallian's Contemporary Abstract Algebra, one of the exercises is to show that if $D$ is a principal ideal domain, then show that every proper ideal of $D$ is contained in a maximal ideal of $D$. My thinking is to just assume for contradiction…
Ebearr
  • 866
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Finding an ideal such that $\mathbb{Z}[x]/I \cong \mathbb{Z}[i]$.

On this released exam, it asks at 2g (slightly modified wording): Give a brief example or show there does not exist an ideal $I$, $I \subseteq \mathbb{Z}[x]$ such that $\mathbb{Z}[x]/I$ is isomorphic to $\mathbb{Z}[i]$ the Gaussian integers. I…
Dair
  • 3,064
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$3 \times 3$ Analogue of Zorn's Vector Matrices

A Zorn vector matrix is a $2 \times 2$ matrix whose diagonal elements are scalars and whose off-diagonal elements are 3-vectors. I read about them on this Wikipedia page for split-octonions. The product of two Zorn matrices is given…
rossng
  • 163
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Is the ideal $(2,3)$ principal in $\mathbb{Q}[x]$?

I believe that I have shown that $(2,3)$ is non-principal in $\mathbb{Z}[x]$. My outline goes something like this: Assume that $(2,3) = (f(x))$ then $f(x)$ divides 2 and 3, that is 2 = $f(x)g(x)$ and 3 = $f(x)q(x)$, so the sum of the degrees of…
Low Scores
  • 4,565
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Question about invariant theory /pre-modern algebra

I am reading an Italian encyclopedia on the history of Invariant Theory. This was an early branch of abstract algebra that was big in the middle of the nineteenth century. Here is a line that I don't understand which I hope that some of you people…
Person
  • 31
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How to prove an element is a unit if and only if the norm is

In the ring $\mathbb{Z}[\sqrt{2}]$, how do I prove that an element $\alpha$ is a unit if and only if $N(\alpha) = 1$? We are told that $N(a+b\sqrt{2}) = a^2-2b^2$. I've shown that $N(\alpha\beta)=N(\alpha)N(\beta)=1$, but since $N(\alpha)…
user230601
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Is $\sqrt[\beta]{\alpha}$ algebraic?

If $\alpha \in \mathbb{C},$ algebraic (over $\mathbb{Q}$) and $ \ \beta \in \mathbb{N} \ $then is $ \sqrt[\beta]{\alpha}$ algebraic? This is my attempt at a proof: Given that $\alpha$ is algebraic then there is $p \in \mathbb{Q}[x]: p(\alpha)=0$ Let…
King Ghidorah
  • 607
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Question about of the polynomial $x^p -x -a$

If $F$ a field with $char(F)=p$. Prove: If $x^p -x -a$ is reducible in $F[x]$ , then this it splits in distinct factors in $F[x]$. I know if for hypothesis $x^p -x -a = P(x)Q(x)$ with $P(x),Q(x) \in F[x]$. Then $deg(P(x))
Angelo
  • 189
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Prove the set $\mathcal{O}_d := \left\{\frac{a + b\sqrt{d}}{2}:a,b \in \mathbb{Z}, a \equiv b \mod 2 \right\}$ is a ring.

Prove that if $d$ is a non-square integer with $d \equiv 1 \mod 4$ then the set $$\mathcal{O}_d := \left\{\frac{a + b\sqrt{d}}{2}:a,b \in \mathbb{Z}, a \equiv b \mod 2 \right\}$$ is a ring, and in particular a integral domain. Little bit stuck…
St Vincent
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