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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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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28 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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2answers
23 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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16 views

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
21 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
33 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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4answers
48 views

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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1answer
22 views

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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25 views

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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1answer
26 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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2answers
36 views

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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2answers
24 views

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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1answer
19 views

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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3answers
69 views

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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0answers
11 views

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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1answer
29 views

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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2answers
39 views

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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2answers
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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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18 views

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
24 views

$H$ 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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42 views

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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42 views

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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1answer
29 views

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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37 views

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
27 views

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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44 views

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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1answer
25 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
22 views

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
31 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
35 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 $\mathbb{Q}$?
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1answer
29 views

The condition that $[LM:K]=[L:K][M:K]$ holds in the field.

Let $L$ and $M$ be intermediate fields of the extension $K\subset F$, of finite dimension over $K$. Assume that $L\cap M=K$ and $[L:K]$ or $[M:K]$ is 2, then $[LM:K]=[L:K][M:K]$ holds. How can I use ...
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Let D be an integral domain with characteristic 3. If x,y are elements of D then (x+y)^3 = x^3 +y^3

I am studying for an exam and the question is: Prove: Let D be an integral domain with characteristic $3$. If $x$ and $y$ are elements of $D$, then $(x+y)^3 = x^3 + y^3$.
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If G is not commutative [on hold]

Edit: Since I did not provide enough detail in my explanation in OP: I have tried to prove this for the general case, but have not come across a suitable proof. I was unsure if I then needed to prove ...
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0answers
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Showing that $\mathbb{Z}_3\times V\simeq\mathbb{Z}_2\times\mathbb{Z}_6$

I'm trying to show that $\mathbb{Z}_3\times V\simeq\mathbb{Z}_2\times\mathbb{Z}_6$, where $V$ is the Klein four-group, $\mathbb{Z}_2\times\mathbb{Z}_2$. I came up with the following method, which ...
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1answer
34 views

Why in this sense this homomorphism is injective?

In this proof:enter link description here Page 19, it gives a construction of outer automorphism of $S_6$,it sends $S_6$ to Perm($S_6$/H)= $S_6$, and by the former proof, it is injective.However, it ...
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3answers
84 views

Confused with Cayley's Theorem in group theory.

Cayley's Theorem: Every group is isomorphic to a group of permutations. $\mathbb Z_6$ is a group and $S_3$ is a permutation, but $\mathbb Z_6$ is not isomorphic to $S_3$. $\mathbb Z_6$ is ...
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1answer
22 views

Proving properties of a subgroup

Let $(G,\cdot)$ be a group, $H \subseteq G, H \neq \emptyset$. Let furthermore $X_{G,H}$ be defined as a construct with the following properties: $X_{G,H}$ is a subgroup of $G$ $X_{G,H} ...
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23 views

Basis of the ring $B=End_R(R^{(\mathbb N)})$

Let $B=End_R(R^{(\mathbb N)})$. Define $u,v \in B$ as $$u(e_{2_{i+1}})=0, u(e_{2_i})=e_i$$$$v(e_{2_{i+1}})=e_i,v(e_{2_i})=0$$ Prove that $\{u,v\}$ is a basis of $B$ as a $B$-module. I've already ...
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2answers
17 views

Showing that a set is a group, and a mapping is a group isomorphism

Let $(G,\cdot)$ be a group, $g \in G$. For $a,b \in G$ define $a * b := a \cdot g^{-1} b$. Show that $(G,*)$ is a group with the neutral element $g$ and $f : (G,*) \rightarrow (G,\cdot), a ...
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1answer
38 views

Show that it is a K-linear map

Let $K \leq K(a)$ a field extension with $[K(a):K]=n$. $K(a)$ is a vector space over $K$. How can I show that a map $\varphi : K(a) \rightarrow K(a)$, $\varphi(e)=ae$, is a K-linear map??
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1answer
25 views

quadratic field extensions of $\mathbb{Q}_p$

Today during class we proved that there were exactly three quadratic field extensions of the $p$-adic number field $\mathbb{Q}_p$. To prove this it was stated that it was enough to look at the group ...
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1answer
21 views

Find the minimum, irreducible polynomial

I have to find the minimum, irreducible polynomial of $$e^{\pi i/3}$$ over $\mathbb{Q}$. I have done the following: $e^{\pi i/3}$ is a root of the equation $x^6-1=0$, right?? ...
2
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1answer
46 views

hom(C) in category theory

I know in the basic definition of a category you have the class hom(C) of morphisms between objects in the category C. What never seems to be clear from textbook definitions is this: Are the members ...