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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Cyclic subgroup of U19

Consider the cyclic subgroup of U19. a. List all it’s generators. b. List all distinct subgroups of U19 and their elements.
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Prove that $S$ is a subring of $\mathbb{Z}_{28}$

Question: $S=\{0,4,8,12,16,20,24\}.$ Prove that $S$ is a subring of $\mathbb{Z}_{28}$ Confusion 1: This might be a dumb question, but when we refer to $[4]$ in $S$, for example, is that the congruent ...
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Discriminant of $x^2+x-1$

I'm working on a homework problem, and am worried I'm going crazy. I believe $x^2+x-1$ is irreducible, but it's discriminant is $1^2-4(1)(-1)=5$ is positive, which would make it reducible (since ...
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Need help to visualise a set (reading Abelian groups)

I am reading Abstract Algebra. I cannot visualise the following example: Let n be a positive integer, and consider the set, $S_n$ of all permutations from the set n = {1, 2, . . . , n} to itself. Let ...
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If $|G|=p^2$ and $p$ is prime then $G\simeq \mathbb Z_{p^2}$ or $G\simeq \mathbb Z_p\times \mathbb Z_p$?

Let $G$ be a finite group of order $|G|=p^2$ where $p$ is a prime. How can I show $G\simeq \mathbb Z_{p^2}$ or $G\simeq \mathbb Z_p\times \mathbb Z_p$? Notice $|G|=p^2$ implies $G$ is abelian ...
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Let $G$ be a group. Prove the equivalence relation: If $H$ is a subgroup of $G$, let $a \sim b$ iff $ab^{-1} \in H$

Let $G$ be a group. Prove the equivalence relation: If $H$ is a subgroup of $G$, let $a \sim b$ iff $ab^{-1} \in H$ To prove an equivalence relation my guess is to show that reflexivity, ...
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When are cancellations allowed in ring?

During the lecture my professored mentioned something like "cancellation is perfectly fine in a ring when dealing with addition, but not with multiplication!". The example he gave was that, in ...
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Order of elements and cosets [on hold]

Show that the order of a coset $|gH|$ divides both $|G/H| =m$ and $|g|=n$ for $H$ is a normal subgroup of $G$. How does one do this. Now Suppose that $|G|=120$ and $|H|=20$. Show that $H$ has all ...
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2answers
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Showing the product of two normal subgroups is normal [on hold]

Prove that if $H$ or $K$ are normal subgroups then $HK=\{hk\mid h\in H,k\in K\}$ is a subgroup. Then if both are normal subgroups, prove that HK is normal.
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Submagmas of natural numbers

What is known about submagmas of natural numbers under addition/multiplication? For example, all subgroups of integers under addition are of the form $~n \mathbb{Z}~$. Are there similar results for ...
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Coset multiplication is not well defined in the case of $S_4$

I have to show that coset multiplication is not well defined in this case. I have to choose 2 cosets $aH$ and $bH$ and locate two different representatives in each coset $a, a' \in aH$ and $b,b' \in ...
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Extending rings

This is a problem I've made up, which I cannot unfortunately solve. Any help will be appreciated. Let $R$ be a commutative ring with unity and Char $R=0$. I want to find the ring $\hat{R}$ ...
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definition of fuzzy translation and fuzzy mutiplication [on hold]

how can a fuzzy subset be a fuzzy translation and fuzzy multiplication in BF/BG-algebra? http://www.indjst.org/index.php/indjst/article/viewFile/37135/29726
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Use of the notation of subgroup vs field extension

Why is it popular to use the idea of subgroups in cases of groups and field extensions in case of fields? In both case one set is the subset of the other along with the restriction of some additional ...
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Find the subgroups of A4

I had a question I was hoping for some help on: Find all of the subgroups of $A_4$ Here is what I know: $A_4$ is the alternating group on 4 letters. That is it is the set of all even ...
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1answer
40 views

Definition of a coproduct and it's universal property - connection?

I have a problem connecting the definition of a coproduct with it's often mentionend universal property. Let's start with the definition (just for two objects): Let $A_1$ and $A_2$ be objects of a ...
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44 views

Localization module is nonzero

Let $R$ be a commutative ring, when $\mathfrak p$ is a prime ideal, there is the localization $M_{\mathfrak p}:=S^{-1}M$, where $S=R\setminus\mathfrak p$. Show: If $M$ is a nonzero $R$-module, ...
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Proving that the sum of elements of two bases is a basis

I am given a (not necessarily orthonormal) basis of a certain finite vector space $\{e_i\}_{1\leq i\leq n}$. Now, after the usual Gram-Schmidt orthonormalization procedure, I end up with an ...
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1answer
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Determine whether a polynomial is primitive in GF(2)[x] [on hold]

Determine whether $ {x^5}+ {x^2}+1$ is primitive in $GF(2)[x]$.
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1answer
22 views

Minimal normal subgroup that is not simple

Let $G$ be a nontrivial finite group. Then $H$ the intersection of all nontrivial normal subgroups has the property that if $K$ is a normal subgroup of $G$ such that $K \leq H$, then $K = H$ or $K$ is ...
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3answers
25 views

Extension field, degree of $[\mathbb Q(i,\sqrt{-3}):\mathbb Q]$

I want to calculate the degree of $[\mathbb Q(i,\sqrt{-3}):\mathbb Q]$, can I do like that: $$X=i+\sqrt{-3}\implies X=i(1+\sqrt{3})\implies X^2=-(1+\sqrt{3})^2\implies X^2=-1-2\sqrt{3}-3\implies ...
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1answer
35 views

An equation which has solution modulo every integer

In the book Abstract Algebra by Dummit and Foote he remarks that there is an equation which has solutions modulo every integer but has no integer solutions. The equation he gives is ...
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What would be an example of a magma such that $x\cdot(x\cdot x)\neq (x\cdot x)\cdot x$?

Let $(M,\cdot)$ be a non-associative magma. What would be an example of $M$ such that there exists $x\in M$ such that $x\cdot(x\cdot x)\neq (x\cdot x)\cdot x$?
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How to directly show that $\mathbb{Z}_{(p)}$ is a local ring with the unique maximal ideal $p \mathbb{Z}_{(p)}$?

I know that $\mathbb{Z}_{(p)}$ is a local ring because it's the localization of $\mathbb{Z}$ over $p$, but is there a direct way to prove that and find its unique maximal ideal? I've been ...
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A concrete example of a unital noncommutative ring without maximal two-sided ideals

Whenever I say ideal in this question I'm talking about two sided-ideals. Does there exist a concrete example of a non-commutative ring with $1$ without maximal ideals? We know that if $R$ is a ...
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Roots Of An Inseparable Polynomial.

Suppose that we have a field $K = \mathbb{Z}_p(t)$ where $p$ is prime. (So this is the field of rational polynomials with $t$ as the variable) Let $f = x^p - t$ be a polynomial in $K[x]$. How can ...
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28 views

Fixed field of two subgroups of $\operatorname{Aut}_{K}{K(x)}$

This link explains $\operatorname{Aut}_{K}{K(x)}$. And I want to know how to solve two problems below in the Hungerford's Algebra, p.256. $7.$ Let $G$ be the subset of $\operatorname{Aut}_{K}{K(x)}$ ...
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If $R$ is a local ring, is $R[[x]]$ (the ring of formal power series) also a local ring?

So, I was trying to find a counter-example that shows not every local ring's lattice of ideals is a chain. I think $F[[x_1,\cdots,x_n]]$ is a good counter-example but I'm not able to show that ...
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Intersection of normal subgroups proof

Show that the intersection of normal subgroups is normal. Let $H_1$ and $H_2$ be normal in $G$, meaning $\forall a \in G$, $aH_1 = H_1a$ and $aH_2 = H_2a$. We show that $a (H_1 \cap H_2) = (H_1 ...
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How to show $\langle a^k\rangle\leq \langle a^\ell\rangle\Leftrightarrow \gcd(\ell, n)\mid \gcd(k, n).$?

Consider the cyclic group $G=\langle a\rangle$ where $o(a)=n$ where $o(a)$ means order of $a$. I'd like to show: $$\langle a^k\rangle\leq \langle a^\ell\rangle\Leftrightarrow \gcd(\ell, n)\mid \gcd(k, ...
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How to prove if $G$ is a group with every non-identity element having order 2 and $H$ is a subgroup, $G/H$ is isomorphic to a subgroup of $G$.

This isn't a homework problem. I'm preparing for an exam, and I have no idea how to solve this problem. Let $G$ is a group such that every non-identity element has order $2$. Let $H$ be a ...
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If $|G|=mp^e$ and $H\leq G$ then $|H|=n p^r$?

If $G$ is a finite group then I may write $|G|=m p^e$ with $p$ prime and $p\not \mid m$. Is it true that if $H\leq G$ then $|H|=n p^r$ with $p\not \mid n$ and $r\leq e$? If the statement above ...
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Integral domain ${\mathbf Z}[\alpha] = \{a + b\alpha\ |\ a,\ b \in {\mathbf Z}\}$

Exercise: Show that the smallest subdomain of complex numbers containing the element $\alpha=\frac{\sqrt{5} - 1}{2}$ is ${\mathbf Z}[\alpha] = \{a + b\alpha\ |\ a,\ b \in {\mathbf Z}\}$. I thought I ...
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How to factorise a number in $\mathbb {Z}[\sqrt {-5}]$?

I am studying quadratic number fields. I have a question about factorization in $\mathbb {Z}[\sqrt {-5}]$ which seems less trivial than factorization in the Gaussian integers. Let $ w=\sqrt {-5} $. ...
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Let $H$ and $K$ be subgroups of a finite group $G$ such that $|H|>\sqrt{|G|}$ and $|K|>\sqrt{|G|}$. Show that $|H\cap K|>1$.

Problem. $|H|>\sqrt{|G|}$ and $|K|>\sqrt{|G|}$ and $H,K\leq G$ where $G$ is a finite group. Prove $|H\cap K|>1$. $$|HK|=\dfrac{|H| |K|}{|H\cap K|}>\dfrac{|G|}{|H\cap K|}\Rightarrow |H\cap ...
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Prove that the number of automorphisms in $\mathbb Q[\alpha]$ equals $1$ $(|Aut\mathbb Q[\alpha]|)=1$

Please, help me to understand this problem: Let $\alpha=\sqrt[3]{2}$ be a root of the polynomial $x^3-2$. a) Prove that the number of automorphisms in $\mathbb Q[\alpha]$ equals $1$ $(|Aut\mathbb ...
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Affine Buildings

I am trying to study affine buildings. So far I learn a lot of theoretical properties and definitions, but it was hard for me to find a good example of this object to "visualize" the theory. (Yes, I ...
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1answer
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$H$ is the smallest subgroup of $S_n$ containing the transposition $(1,2)$ and the cycle $(1,2,…,n)$ [on hold]

For $n>2$, if $H$ is the smallest subgroup of $S_n$ containing the transposition $(1,2)$ and the cycle $(1,2,...,n)$ , then (A) $H = S_n$ (B) $H$ is abelian (C) The index of $H$ in $S_n$ ...
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Let $|G|=17.\;$ How many non-isomorphic subgroups of G are there? [on hold]

I don't know how to find non-isomorphic subgroups of a group. Thanks.
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Group objects in category of $\mathcal{Set}$ are groups - How to proof?

Reading about group objects in categories, it's a fact that a group object is in the category of $\mathcal{Set}$ just a common group. I am trying to give an actual proof of this, but I'm a bit ...
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Equivalent definitions of an algebra over a ring

I have seen two different definitions from several sources of an algebra over a ring. From Wikipedia: Let R be a commutative ring. An R-algebra is an R-Module A together with a binary operation ...
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Prove that $\langle a^n \rangle \bigcap \langle a^k \rangle = \langle a^{lcm (n,k)} \rangle$

Let $G$ be a group. Let $a$ be an element. Let $n,k$ be pozitive integers. Let $m$ be least common multiple of $n$ and $k$. Prove $\langle a^n \rangle \bigcap \langle a^k \rangle = \langle a^{m} ...
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1answer
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No sub-integral domain of Z with prime characteristic?

I try to find a subring of Z which it is integral domain and characteristic is a prime. Until now, I can't find it. But i believe that this proposition is true. Please help me prove or disprove.
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Give an example of an infinite non-commutative ring R with char(R)=15 [on hold]

I know I need to use matrices, but I'm not sure how. I know matrices are non-commutative however I'm confused about the characteristic part.
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31 views

Suppose that $L:K$ is algebraic. Show that the following are equivalent:

$(A)$ $L:K$ is normal $(B)$ if $j$ is any monomorphism from $L$ to $\overline{L}$ which fixes $K$ then $j(L) \subseteq L$ $(C)$ if $j$ is any monomorphism from $L$ to $\overline{L}$ which fixes $K$ ...
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1answer
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Proof verification for $fgh=1_A\dots\implies f,g,h$ are all bijections. - Cohn - Classic Algebra Page 15

Is the proof below correct? Thank you for your time! Notation: $xfgh\equiv h(g(f(x)))= (h \circ g \circ f)(x)$ Theorem: If $f:A\to B, g:B\to C, h:C\to A$ are three mappings such that $fgh=1_A$, ...
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1answer
33 views

General way to solve equations of congruent classes?

So I'm taking my first abstract algebra class, and I have some homework like "Solve the equation $x^2 + [3]\times x+[2]=[0]$ in $\mathbb{Z}_6$. This doesn't seem to difficult to do since I can easily ...
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2answers
52 views

Intuitive Meaning of Quotient Ring

I have been trying to understand the intuitive meaning of quotient ring for quite some times but unfortunately I could not find any. Now I am approaching this question from a different angle, hoping ...
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1answer
38 views

What does $[5^{2000}]$ equal to in $\mathbb{Z}_4$?

Question: What does $[5^{2000}]$ equal to in $\mathbb{Z}_4$? My proof: (which I doubt whether its correct or not since it doesn't use the hint in the book) $[5^{2000}]=([5])^{2000}$ Since $5 \equiv ...
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Galois group of $x^{2^k}+1$

What is the Galois group of $f(x)=x^{2^k}+1$ over $\mathbb{Q}$?