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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Verify that R is a ring

Let $\alpha = \frac{1}{2}(1+\sqrt{-19}) \in \mathbb{C}$ and $R = \{a+b\alpha\mid a,b \in \mathbb{Z}\} \subseteq \mathbb{C}$. Is R an integral domain with unity? My attempt: (Please correct me if I ...
2
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2answers
70 views

Prove that $a+ib$ is prime in $\Bbb Z[i]$, of $a^2+b^2$ is prime in $\Bbb Z$.

Prove that $a+ib$ is prime in $\Bbb Z[i]$, of $a^2+b^2$ is prime in $\Bbb Z$. My Try: We can easily show that $\Bbb Z[i]$ is a FD but how can we show that $\Bbb Z[i]$ is a UFD. Because if we can show ...
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+50

Let $H$ be a subgroup of the group $(R, +)$ such that $H$ $∩$ [-1,1] is a finite set containing a non zero element. Show that $H$ is cyclic.

Observations: Since $H$ is a subgroup of $(R, +)$ so $0 \in H.$ If $1 \in H,$ then all positive integers belong to $H.$ But $H$ is closed wrt addition, so the negative integers must belong to $H$ ...
4
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2answers
67 views

How to show that $3\mathbb{Z}/15\mathbb{Z} \cong \mathbb{Z}/5\mathbb{Z}$?

How to show that $3\mathbb{Z}/15\mathbb{Z} \cong \mathbb{Z}/5\mathbb{Z}$ as $\mathbb{Z}$-module over $\mathbb{Z}$? My proof: Define surjective function $f:3\mathbb{Z}/15\mathbb{Z} \rightarrow ...
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0answers
24 views

Module product and coproduct

Completely lost when reading this: "Defintion: If the index set $T$ is finite, the coproduct and product are the same module. If $T$ is infinite, the coproduct or sum $$\coprod_{t\in ...
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2answers
91 views

Cohen-Macaulay and regular rings

I know this is a simple question but to make sure....: $A$ is a commutative ring which is Cohen-Macaulay and for every maximal ideal $\mathfrak{m}$ in $A$ if $\dim A_{\mathfrak{m}}=\dim A$ then ...
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37 views

Inverse property for groups Proof

I was wondering if (1) this proof is correct, and (2) if other proofs exist for the following: Prove that $(a_1a_2...a_n)^{-1}=a_n^{-1}a_{n-1}^{-1}...a_1^{-1}$ where $a_i \in $ a Group $G$ Proof by ...
3
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0answers
26 views

Image of homomorphism with kernel $N$ isomophic to $M/N$

Let $M$ be an $R$-module and $N$ be a submodule. Let $f:M\rightarrow M'$ be a surjective homomorphism with kernel $N$. How to show that $M'$ is isomorphic to $M/N$? My thought: Define ...
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35 views

Simple composition factor

We know that if $e$ is a primitive idempotent in a semiperfect ring then $Re/Je$ is a simple left $R$-module, where $J$ is the Jacobson radical of $R$. Now, suppose that $e$ is an idempotent in an ...
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1answer
25 views

To show module homomorphism being injective

Suppose $R$ is a field, $M$ is a right $R$ module, and $f : R_R \rightarrow M$ is a non-zero homomorphism. Show that $f$ is injective. My work: The homomorphism is uniquely determined by $f(1)$. If ...
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1answer
31 views

given polynomial has a root in $Z_p$…

To check $f_x$ is irrudicble or not in $f_p$ check wheather 0,1,2 , p-1 is a root of $f_x$...if any of this is a root then it is not irrudicible.. is this method applicable here?
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3answers
33 views

Prove that there is a fixed point in any subgroup $H$ of $S_4$ of order $6$.

Prove that in every subgroup $H$ of $S_4$ of order 6 there is a fixed point in {$1,2,3,4$}, i.e, there exists $1\le i\le 4$ such that $h(i)=i$ $\forall h\in H$. $Start$: Suppose there is a subgroup ...
2
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0answers
49 views

Isomorphisms between $(\mathbb Z_4,+)$ and $(U_5,*)$

So I am asked to find all the isomorphisms between $G = (\mathbb Z_4,+)$ and $H = (U_5,*)$. I solved it as follows: we will have two isomorphism corresponding to the two generators of $U_5$. The ...
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2answers
43 views

Show that $2$ is not prime in $\mathbb Z[\sqrt{-d}]$ for odd prime $d$

I have to show that if $d > 2$ is a prime number, then $2$ is not prime in $\mathbb Z[\sqrt{-d}]$. The case when $d$ is of the form $4k+1$ for integer $k$ is quite easy: $2\mid ...
2
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1answer
51 views

Find $n$ s.t. $Aut(\mathbb Z_2 \oplus \mathbb Z_4 \oplus \mathbb Z_4 \oplus \mathbb Z_6) \cong U(n).$

can we determine the automorphism group of a $U$-group i.e $Aut(U(n))$ ? I need to find $n$ s.t $Aut(\mathbb Z_2 \oplus \mathbb Z_4 \oplus \mathbb Z_4 \oplus \mathbb Z_6) \cong U(n).$ ? I started by ...
3
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6answers
143 views

Prove that $x-1$ is a factor of $x^n-1$

Prove that $x-1$ is a factor of $x^n-1$. My problem: I already proved it by factor theorem† and by simply dividing them. I need another approach to prove it. Is there any other third ...
2
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1answer
59 views

Rings in which every maximal ideal is a direct sum of cyclic modules

Let $R$ be a ring in which every maximal ideal is a direct sum of cyclic $R$-modules. Now let $I$ be a proper ideal of $R$. What is the structure of $I$? Is it true that $I$ is a direct sum of ...
2
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2answers
90 views

prove or disprove $H$ is a subgroup

If $H$ is a nonempty subset of a group $G,$ and if $a,b\in H,$ then $a^{-1}b^{-1} \in H,$ can we prove that $H$ is a subgroup of $G$? if not, how to disprove it?
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1answer
48 views

What are the cosets of this presentation?

I'm reading a book on algebra, and they give a presentation for $S_3$, with 6 elements $\{1, x, x^2, y, x y, x^2y\}$ as $$x^3 = 1,\quad y^2 = 1,\quad y x=x^2y$$ Now later in the book, there is a ...
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1answer
31 views

How many possible isomorphisms do we have between G and H? [duplicate]

Let $G=(Z_4,+)$ and let $H=(U_5,*)$ where $U_5 = \{[1],[2],[3],[4] \}$ . I know that $[1]$ and $[3]$ are both generators for $G$. I also know that $[2]$ and $[3]$ are both generators for $H$. In order ...
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23 views

The abelianness of the quotient group of an abelian group.

I am working on an assignment for my abstract algebra class. The question states: Let $A$ be an abelian group and let $B$ be a subgroup of $A$. Prove that $A/B$ is abelian. I was under the ...
3
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2answers
244 views

Example of non-Abelian group with 4, 5, or 6 elements of order 2

Is there any example for non-Abelian group in which has more than $3$ elements of order $ 2$, but has less than 7 elements of order $2$?
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0answers
30 views

Is my observation correct regarding restriction of scalars?

Let $\alpha: \Lambda\to \Gamma$ be a ring homomorphism, then $ _\Lambda\Gamma_\Gamma$ is a bimodule. We have the following pairs of adjoint functors $$ \mathbf{Mod_\Lambda} \xrightarrow{\cdot\; ...
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2answers
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Finding an order of a coset in $A/B$ where $A$ is a free abelian group and $B$ is a subgroup.

Let $A$ be a free abelian group with basis $x_1,x_2,x_3$ and let $B$ be a subgroup of A generated by $x_1+x_2+4x_3, 2x_1-x_1+2x_3$. In the group $A/B$ find the order of the coset $(x_1+2x_3)+B$. How ...
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0answers
22 views

Find up to isomorphism all the quotient groups of composition series of a group of order $30$.

I can't seem to understand what I should do here... All I did so far is proving that $G$, (such a group), is not simple. But there are many cases, I can't really tell what $n_2,n_3$ and $n_5$ are, ...
2
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1answer
47 views

Is there a formalism for a universal mathematical representation of algorithms?

I don't know if my question is correct so excuse me if I'm not 100% clear about what I would want to know. Is there a formalism which can capture all possible algorithms (mathematically speaking) ? ...
2
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0answers
62 views

Image sheaf is the sheafification of the image presheaf

This is an exercise in Vakil's notes on foundations of algebraic geometry. Suppose $\Phi:\mathscr{F}\to\mathscr{G}$ is a morphism of sheaves of abelian groups, show that the image sheaf Im $\Phi$ ...
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1answer
19 views

Group Z, find<m,n>

How to find <8,14>, in group Z under addition, Any integer k such that the subgroup is . For 8, <8>={8n : n$\in$Z}, <14>={14n:n$\in$Z}, so <8,14>={22n and 6n:n$\in$Z}, is it right? how to ...
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3answers
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Are the $2\times 2$ symmetric matrices a ring?

Ok so I am looking at Rings. I saw somewhere that the $2 \times 2$ symmetric matrices with entries in $\mathbb{R}$ is a ring. But if we look at matrix multiplication I am not convinced: If $ A = ...
2
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2answers
54 views

How many homomorphisms are there from $\Bbb{Z}_6 \to \Bbb{Z}_{18}$?

I need to determine how many homomorphisms there are from $\Bbb{Z}_6 \to \Bbb{Z}_{18}$. I have never solved that kind of question. I do know that orders are preserved and that some elements can be ...
3
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1answer
44 views

Irreducible representations of $S_3$

I am trying to do a problem from artin's algebra 2nd ed (Chapter 10, Exercise 2.3) but having trouble: Let $(\rho , V)$ be a representation of the symmetric group $S_3$. Let $x=(123), y=(12)$ be the ...
4
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1answer
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Finitely Many Extensions of Fixed Degree of a Local Field

How does one show that there are only finitely many degree $n$ extensions of a local field? I understand how this follows from class field theory in the Abelian case but don't understand how to do the ...
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Multiple products of submodules

NOTE: This is part of a homework, so only worry about the question regarding notation. We have the following conditions: $R=\mathbb{Z}$, $I = \mathbb{Z}_{>0}$, and $M_i = \mathbb{Z} / i ...
4
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2answers
95 views

In a Noetherian integral domain, a principal prime ideal can't have proper non-zero prime ideals

Let $R$ be an integral domain and Noetherian. Let $P \subset R$ be a non zero prime ideal. Prove that if $P$ is principal then there is no prime ideal $Q$ such that $0 \subsetneq Q \subsetneq P$. ...
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Algebra A over a field F contains no non-trivial left F-ideals if and only if A contains no non-trivial right F-ideals [on hold]

Algebra $A$ over a field $F$ contains no non-trivial left $F$-ideals if and only if $A$ contains no non-trivial right $F$-ideals. Why this fact is true? Or is it true? I think it's easy thing, ...
3
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2answers
45 views

Find order of polynomials in a group [on hold]

$G$ is the group of polynomials under addition with coefficient from $\mathbb Z_{10}$. How to find the order of $f(x)=7x^2 + 5x+4$? If $h(x)=a_nx^n+...+a_0 \in G$, how to get the order of $h(x)$ if ...
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1answer
54 views

Automorphism group of the ring $\mathbb{F}_3\left[t,\frac{1}{t}\right]$

Let $R=\mathbb{F}_3\left[t,\frac{1}{t}\right]$ be a ring. What is the simplest form of $\mathrm{Aut}(R)$ ? Here $t$ is a variable and $R$ is the smallest ring contained in field ...
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1answer
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239 views

What is a Simple Group?

I am working on this problem from class note: Let $G'$ be a group and let $\phi$ be a homomorphism from $G$ to $G'$. Assume that $G$ is simple, that $|G| \neq 2$, and that $G'$ has a normal ...
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1answer
23 views

All automorphisms of splitting fields

Let $M \le \mathbb{C} $ be the splitting field of polynomial $ f(x) \in \mathbb{Q}[x] $. Find all automorphisms of field $ M $ in cases: 1) $ f(x) = x^6 - 1 $ 2) $ f(x) = x^{2011} - 1 $ 1) In ...
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1answer
37 views

Galois group of $X^5-1\in\mathbb F_7$

I want to find the Galois group of $X^5-1$ over the finite field $\mathbb F_7$ but I don't know how to find Galois groups over finite fields. Over $\mathbb Q$ the Galois group $\text{Gal}(\mathbb ...
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3answers
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Simple form of a ring

What is a simple form of this ring: $$\mathbb{Z}[\sqrt{2}][x]/(5,x^2+1),$$ I know that $\mathbb{Z}[\sqrt{2}][x]=\mathbb{Z}[x,y]/(y^2-2)$. Probably, I should use second theorem of isomorphism, but I ...
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1answer
41 views

Elliptic curve-point at infinity

In my lecture notes we have the following: $$P \oplus Q \oplus R =O \Leftrightarrow P, Q, R \text{ are collinear }$$ So $$P \oplus Q \oplus O =O \Leftrightarrow Q=-P$$ that means that $Q=-P$ ...
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Criteria for the existence of zero-divisors and idempotent elements in the integers modulo $n$

I need help in establishing or at least deciding the validity of the following two criteria: There are in the ring $Z_n$ non-trivial zero divisors if only if $n$ is divisible with some square. ...
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2answers
39 views

Relationship between submonoids and subgroups

I'm taking my first abstract Algebra course. During one of the last lessons, my teacher told us that If $G$ is a group and $M$ is a finite submonoid of $G$, then $M$ is a subgroup of $G$. For the ...
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quick question on Galois theory

I am reading a book on Galois theory and, as per usual (why is that?), all sorts of unproven properties start to magically appear in the proofs of the couple of theorems that really matter. The author ...
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52 views

question regarding group theory proof

Can someone please explain the sentence in red?, how does it follow?
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33 views

Bigenetic properties of finite group [on hold]

Nilpotency, supersolubility and polycyclicity are bigenetic properties of the class of all finite group. Let be : P is property, X be a class o group. We say that P is a bigenetic property of ...
3
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2answers
81 views

Order of any element divides the largest order.

Let $A$ be a finite Abelian group and let $k$ be the largest order of elements in A. Prove that the order of every element divides $k$. This is my attempt, I sense there is something wrong\incorrect ...
4
votes
2answers
101 views

Abstract Algebra in analyzing computer science

I would like to know of some uses of algebraic structures to study computer science. Parallels of what I am looking for would be stuff like the fundamental group/homology/cohomology in topology and ...