Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, and other advanced topics.

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Centralizer of $A_m$ in $A_n$

Problem: I want to calculate centralizer of $A_m$ in $A_n$. As far as I know, $A_{i},\ i\geq 5$ is simple. So if $m\geq 5$, then $$ C := \{ a\in A_n|\ ax=xa\ \forall x\in A_m\} = A_{n-m} $$ ...
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16 views

Order of an Element Under a Homomorphism

I was reading an example about describing homomorphisms and I'm having a bit of trouble with this one. Show there is no group homomorphism $f : \mathbb{Z}_{10} \to \mathbb{Z}_{25}$ such that $f(1) = ...
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22 views

Polynomial in two variables

In the ring of polynomials of two variables $\mathbb{C}[x,y]$, the polynomial $f(x,y)=x^2+y^2+1$ is irreducible ?
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$H\trianglelefteq A$ and $K\trianglelefteq B\Rightarrow HK\trianglelefteq AB$?

Let $G$ be a group and $A, B\leq G$. Suppose $H\trianglelefteq A$ and $K\trianglelefteq B$. Is it true that $HK\trianglelefteq AB$? Notation: $\leq$ means subgroup and $\trianglelefteq$ means normal ...
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non-zero elements in $\mathbb Z_3[i]$ form an abelian group

How shall I show that all non-zero element of $\mathbb Z_3[i]$ form an abelian group of group of order $8$ under multiplication... Please any hint how shall I show this result?
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$\mathbb Z[\sqrt d]$ is not a field.

How shall I show that $\mathbb Z[\sqrt d]=\{a+b\sqrt d~~\big|a,b\in \mathbb Z\}$ is not a field? It is an integral domain, so the thing it lacks maybe is every element does not have a ...
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1answer
30 views

Equivalence of categories is an equivalence relation

Suppose that $B \xrightarrow{F}C$ and $C \xrightarrow{G}D$ are equivalences of categories. I want to show that $G \circ F$ is an equivalence. (This becomes easy if I use that a functor is an ...
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61 views

prove that $ab=1$ implies $ba=1$.

I have a doubt how to prove: If the product of any pair of non-zero elements of $R$ is non-zero , prove that $ab=1$ implies $ba=1$. how shall I make use of the fact : product of any pair of ...
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18 views

Canonical Structure to figure out proof strategy

Even though the integers form a Euclidean domain, most results about can be derived from the weaker fact that it is a PID (I do realize that establishing it forms a PID uses the fact that it is an ED, ...
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38 views

Multiplication of subgroups

Let $H$ and $K$ be subgroups of a finite group $G$. Define $HK = \{hk\mid h \in H, k \in K\}$ and $KH = \{kh\mid k \in K, h \in H\}$. a) Show that in general $HK \ne KH$. (For example, consider $G = ...
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43 views

Representation is reducible

Suppose $V$ is a representation of a finite group $G$ over a field $k$ of characteristic $0$, and suppose dim$V=3$ and $\wedge ^2V$ is reducible. Then $V$ is reducible. I was trying to do it by ...
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20 views

Center of a Group and automorphisms

The center of a group, G is defined as: Z = {a|ag = ga for all g ∈ G}. Let $\phi$ be an automorphism of a finite group G to G. Show that for any a ∈ Z then $\phi(a) $∈ Z. Conclude that $\phi(Z) = Z$. ...
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1answer
30 views

Zero divisors in $(\mathbb Z_n,+,*)$

How to understand this : An element $a$ in $(\mathbb Z_n,,+,*)$ is a zero divisor iff $a$ and $n$ aren't coprime... EDIT: Is it also true that an element a in $(\mathbb Z_n,,+,*)$ is a unit iff ...
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$\mathbb R\oplus \mathbb R$ is an Integral Domain or a Division Ring?

Can anyone help me with these two doubts of mine: Is the ring $\mathbb R\oplus \mathbb R$ an Integral Domain or a Division Ring? My notes state that the ring of Gaussian integers(i.e. $\mathbb ...
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1answer
37 views

What is a quick proof that $f \in \mathbb{C}[X_1,\dotsc,X_n]$ is determined by its induced function on $\mathbb{C}^n$?

For $f \in \mathbb{C}[X_1, \dotsc, X_n]$, we have the induced function $\bar{f}: \mathbb{C}^n \to \mathbb{C}$ given by evaluation. The association $f \mapsto \bar{f}$ is injective. Is there a quick ...
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107 views

Is product of prime ideals prime?

I'm trying to show that the product of ideals $(x_1, x_3)$ and $(x_2, x_4)$ in $\mathbb C[x_1, x_2, x_3, x_4]$ is a radical ideal, but no other way that I can think of works. So, is the product ...
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114 views

Questions related to maximal ideals

In my previous sessional exams, I was asked to prove these two: 1) Find a ring which doesn't have a maximal Ideal. 2) If a ring has unity, then it has a maximal Ideal. About the ...
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40 views

Find the kernel of the group homomorphism $G\to\text{Bij}(G/H),\;a\mapsto(b\mapsto abH)$

Let $G$ be a group and $H\subset G$ be a subgroup. Find the kernel of the group homomorphism $$G\to\text{Bij}(G/H),\;a\mapsto(b\mapsto abH)$$
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24 views

Suppose that a function $f: G \to H$ is an isomorphism? Must f inverse be an isomorphism as well?

Suppose that a function $f: G \to H$ is an isomorphism? Must f inverse be an isomorphism as well? Prove it if it is so, else, provide a counterexample. I think that it is, the following is my proof: ...
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34 views

Showing that an ideal is prime

I think that $k[x,y,z]/(z-1, x^2-y)$ can be identified as a subset of $k[x,y]$ with all polynomials whose $x$ terms are only degree one. Therefore I conclude that $k[x,y,z]/(z-1, x^2-y)$ is integral ...
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1answer
29 views

Does the dimension of the row space equal dimension of the column space for complex matrices?

In the case of real matrices, the dimension of the row space equals the dimension of the column space: $\text{dim } \text{span}(A) = \text{dim }\text{span}(A^T)$. But for complex matrices the ...
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1answer
32 views

no. of elements in $\mathbb Z[i]/\langle 3+i\rangle$. [duplicate]

$\mathbb Z[i]/\langle 3+i\rangle$ can be represented as :$\{a+3b+\langle 3+i\rangle\big|~~a,b\in \mathbb Z\}$ How shall I find the total no. of elements in $\mathbb Z[i]/\langle 3+i\rangle$.. ...
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What is so special about Klein 4-group?

This is my first course in abstract algebra and so far I am only learning about groups. So is there anyone who can explain to me why Klein 4-group is so special that it warrants a category of its own. ...
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1answer
16 views

Universal property for vector spaces

QUESTION: I would like some hints for part b.I am unsure of how to show that $\widetilde{T}$ is surjective or injective for that matter. My construction for $\widetilde{T}$ is ...
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66 views

Follow-up regarding $f(x,y)\cdot f(y,z) = f(x,z)$

I posted this question about a year ago. And I started thinking about it again. The question as follows: Under what conditions does $f(x,y) \cdot f(y,z) = f(x,z)$ not imply that $f(x,y)$ can be ...
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49 views

Insights on coset in abstract algebra

I am taking course on abstract algebra. I found the concept of coset is very important. However, I have not found any insights on my textbook (and by myself). I believe there must be great example on ...
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1answer
37 views

Show that the element $z=i \cos \frac{\pi}{3}+\sin \frac{\pi}{3} = i( \cos \frac{\pi}{3} - i \sin \frac{\pi}{3})$ belongs to $U_{12}$

Show that the element $\displaystyle z=i \cos \frac{\pi}{3}+\sin \frac{\pi}{3} = i( \cos \frac{\pi}{3} - i \sin \frac{\pi}{3})$ belongs to U12 What I don't understand: In what way $i \cos ...
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Lucas' Theorem for $p$-adic integers?

Does Lucas' Theorem hold for $p$-adic integers? More specifically, does it specifically hold for the case that, given a $p$-adic integer: $$x = x_0 + x_1p + x_2p^2 + \cdots = \sum_{i=0}^\infty ...
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1answer
30 views

Show that $a \star b=a \cdot b+a+b$ is binary operation for the group $\Bbb{ Q} - \{-1\}$

The group $\left(\Bbb{ Q} - \{-1\},\star\right)$ has as its underlying set the rational numbers different from $-1$ and the operation $\star$ is defined as $a \star b=a \cdot b+a+b$ where ...
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1answer
64 views

Constructing a colimit

Consider a functor $F: \mathcal{I} \rightarrow \textrm{Sets}$ where $\mathcal{I}$ is small. Then the colimit of $F$ is given by $\amalg_{i \in \textrm{Ob}(\mathcal{I})}F(i)/{\sim}$ where $\sim$ is the ...
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26 views

Something similar to Euler's theorem

If $p$, $q$ are not equal primes. $n=pq$, $\varphi(n) = (p − 1)(q − 1)$, $d = \gcd(p − 1, q − 1)$. Is it true that for any $a$ such that $\gcd(a, n) = 1$ holds $a^{\frac{\varphi(n)}{d}} \equiv 1 ...
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1answer
34 views

Is there a basis for the continuous functions space?

I've been searching all over the Internet for this but without finding a satisfying answer. This might be a dumb question, but I would like to know the answer anyway. Is there a set of continuous ...
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The diophantine equation $A^2+B^2=C^2$ for integer-valued polynomials

How can I find the solutions to this diophantine equation in $\Bbb{Z}[X]$: $$A^2+B^2=C^2 \, ?$$ Here $A$, $B$, $C$ are polynomials.
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Proving commutativity [duplicate]

Let $R$ be a ring in which $x^2=x$ for all $x\in R$ where $x^2$ of course denotes $x\cdot x$. a. prove that $x+x=0$, for all $x \in R$ b. prove that $R$ is commutative. I have done part a but how ...
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Is abstract algebra (mostly?) restricted to $2$-ary operators?

This may be due to my own pure ignorance but it's my experience that all abstract algebra I've been introduced to, both in actual courses and in self-studies only exclusively deals with algebraic ...
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57 views

Logic problem: Atiyah-Macdonald 1.11

Proposition 1.11 in Atiyah-Macdonald's "Introduction to commutative algebra" states the following: "Given an ideal $I$ in a ring $A$ and $p_1, \dots p_n$ prime ideals, then $I \subset \cup_i p_i$ ...
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Internal Direct Sum [on hold]

Let F be a field and V a vector space over F. Let $W_1$, $W_2$, U be proper subspaces of V. Assume that each $W_i$ is a complement of U, that is $W_i \oplus U=V$ is an internal direct sum. Do not ...
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1answer
25 views

Permutations Isomorphic to $S_4$

Prove that the group generated by permutations $(0 2 6 4)(1 3 7 5)$ and $(4 2 1)(6 3 5)$ are isomorphic to the symmetric group $S_4$. I approached this problem by labeling the vertices of a cube. ...
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1answer
16 views

Equivalence for binary quadratic forms with positive square discriminant

I recently encountered an interesting proposition without proof: If $f(x,y)$ is a quadratic form whose discriminant is a non-zero perfect square, then $f(x,y)$ is equivalent to a form $a*x^{2} + ...
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Find a non-trivial semidirect decomposition of the following groups

Find a non-trivial semidirect decomposition of the groups $S_n$, $n \geq 3$, $D_{2n}$, $n \geq 3$ and $A_4$. Prove that $A_n$, $n \geq 5$ and $Q_8$ have no non-trivial semidirect decompositions. How ...
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1answer
10 views

torsion of a module and exactness

Given a PID $A$ and $A$-modules $M$, $M'$, and $M''$. Assume that $$0\rightarrow M'\xrightarrow{f} M\xrightarrow{g} M''$$ is an exact sequence then prove that $$0\rightarrow ...
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1answer
21 views

$A=\{(3x,y)\mid x,y\in \mathbb Z\}$ is a maximal ideal of $\mathbb Z \oplus \mathbb Z$

I've a question from gallian which states: Show that $A=\{(3x,y)\mid x,y\in \mathbb Z\}$ is a maximal ideal of $\mathbb Z \oplus \mathbb Z$.Generalize.What happens if $3x$ is replaced by $4x$... ...
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61 views

$\mathbb R[x]/\langle x^2+1\rangle$ is a field

How to show that $\mathbb R[x]/\langle x^2+1\rangle$ is a field. I wrote the representation of $\mathbb R[x]/\langle x^2+1\rangle$ =$\{a+bx+\langle x^2+1\rangle\big|a,b\in \mathbb R\}$. Now how ...
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2answers
30 views

Units and nilpotents in quotient ring. [on hold]

$A$ is a commutative ring and $N(A)$ is the nilradical of $A$. If $A/N(A)$ is a field, show that every $a \in A$ is invertible or nilpotent.
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Krull dimension of a finitely generated integral domain over $k$ is equal to the transcendence degree.

This theorem is from Matsumura (p.34) Let $k$ be a field and $A$ an integral domain which is finitely generated over $k$. Then $\dim A = \operatorname{trdeg}_k A$ (where $\operatorname{trdeg}_k ...
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1answer
20 views

Homomorphism between 2 abelian groups sending one given element to another given element

Let $G$ and $G'$ be arbitrary abelian groups. Fix a $g \in G$ and $h \in G'$. Then does there exist a homomorphism $\phi$ such that $\phi(g) = h$?
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34 views

Doubt regarding zero elements in factor ring :$\mathbb Z[i]/\langle3-i\rangle$

I have the factor ring $\mathbb Z[i]/\langle3-i\rangle$ and am asked to find elements zero in this ,they are $0,3-i,i(3-i),(3-i)+i(3-i)$. But I can't understand how do we guarantee these are the only ...
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3answers
59 views

Proving ring $R$ with unity is commutative if $(xy)^2 = x^2y^2$

I tried several methods to solve this but couldn't get through. Now the solution in almost all the textbooks goes like this. First take $x$ and $y+1$ so that $ (x(y+1))^2 = x^2(y+1)^2 => xyx = ...
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1answer
51 views

What is meant by a kernel and homomorphism in algebraic structures?

I have just started with discrete maths. I was doing some group theory and I stumbled upon kernel and homomorphism. I didn't understand what was written in the book. I googled it up and also looked ...
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35 views

Tensor product of R-algebras

I'm dealing for the first time with tensor product of $R$-algebras $S,T$ (where $R$ is a commutative ring). $S\otimes_{R}T$ Can someone explain me what is the advantage given by the fact that $S$ ...