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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Good problem book on Abstract Algebra

I am currently self-studying abstract algebra from Artin. In that background, I am looking for a problem book in a spirit somewhat similar to Problems in Mathematical Analysis by AMS so that I have a ...
6
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1answer
67 views

When does a formula for the roots of a polynomial exist?

My question is straightforward to pose: given a polynomial $f$ over a subfield of $\mathbb{C}$, are there conditions which guarantee the existence of a closed formula for the roots of $f$ in terms of ...
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8answers
364 views

Every infinite abelian group has at least one element of infinite order?

Is the statement true? Every infinite abelian group has at least one element of infinite order. I am searching for an infinite abelian group with all elements having finite order. Please help me ...
3
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2answers
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Find the number of subgroups in $Z_p \times Z_p \times Z_p$

let $p$ be a prime number ; I want to find the number of subgroups in $G = Z_p \times Z_p \times Z_p$ $(Z_p = \mathbb{Z}/p\mathbb{Z})$. I know that there is $p^2 + p + 1$ copies of $Z_p$ in $G$ for ...
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1answer
45 views

Automorphisms in $Z_n$

I know an Automorphism is a group G that is an isomorphism where $h:G \rightarrow G$, (G is being mapped to itself) and that Aut(G) is the set of all Automorphisms in G. I was wondering how I would ...
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0answers
39 views

Show for $x\in G$ written as product of $hk$.

Show for $x\in G$ written as product of $hk$ for $h \in H$ and $k\in K$. Let $G$ be of order $p^km$ for $p$ is prime and does not divide $m$. $H=(x\in G\mid x^{p^k}=e)$ and $K=(x\mid x^m=e)$. G is not ...
0
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2answers
85 views

Choosing “Conjugating element” from a subgroup

Let $H\leq G$ and $N\unlhd G$ such that $G=HN$ and $H\cap N= (1)$. My question is Prove that if two elements of $H$ are conjugate in $G$, then they are conjugate in $H$. What i have done so far ...
3
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2answers
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If $H\cap K=e$ then $H$ and $K$ are normal

For $|G|=p^km$ for $p$ is prime and $p$ does not divide $m$. Let $H=[x\in G \mid x^{p^k}=e]$ for $H<G$ and let $K=[x\mid x^m=e] $ for $K<G$. We are not assuming G is abelian Show that $H\cap ...
6
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2answers
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Infinite Units for $\mathbb{Z}[\sqrt{7}]$

Suppose that $\alpha \in \mathbb{Z}[\sqrt{7}]$ where $\alpha$ is of the form $a + b\sqrt{7}$ where $a, b \in \mathbb{Z}$. Because $\alpha$ is a unit if and only if $N(\alpha)=\pm 1$ we must show: ...
4
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1answer
51 views

If a nilpotent group $A$ act on a group $G$ then $G$ is solvable.

Theorem: If a nilpotent group $A$ act on $G$ by automorphism and $C_G(A)=e$ then $G$ is solvable. I hope that the statement of theorem is true. It should belong to Hartley, but when I googled it I ...
3
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2answers
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The interpretation of ideals of a ring.

Ideals of a commutative ring (I have only studied the commutative case) are thought of as generalized numbers (in algebraic number theory) and as ring homomorphisms (through the ideal as kernel ...
2
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2answers
35 views

$R$ is a ring. Prove that $R/(0_R)\cong R$

$R$ is a ring. Prove that $R/(0_R)\cong R$. I don't quite understand what $R/(0_R)$ looks like. By definition of quotient ring, it should have cosets $a+(0_R)$ where $a\in R$. So $R/(0_R)$ and ...
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2answers
56 views

Infinite algebraic extension of a finite field

I have recently started studying algebraic field extensions and I got to know that algebraic closures $\overline{F}$ of finite fields $F$ are infinite. Therefore, I've asked myself the following ...
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2answers
42 views

Why $P_1\neq P_1P_2$?

Question: If $P_1,P_2$ are distinct prime ideals of an artinian ring, why is it that $P_1\neq P_1P_2$? I know that prime ideals of an artinian ring are maximal, but still, I can't see why ...
3
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2answers
64 views

Show that $\mathbb Q(\sqrt p) \not\simeq\mathbb Q(\sqrt q)$

I'd like to show that for $p,q$ distinct primes, the extensions $\mathbb Q(\sqrt p),\mathbb Q(\sqrt q)$ are not isomorphic. I don't really have knowledge of the "high-level language" of algebraic ...
2
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0answers
41 views

Is this article about exponential wrong?

Wikipedia : http://en.wikipedia.org/wiki/Formal_power_series Assume that the ring R has characteristic 0. If we denote by exp(X) the formal power series $exp(X)=1+X+\frac{X^2}{2} + ...
2
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1answer
54 views

For $G$ an abelian group and $H$ a subgroup, is $[G : H]$ the smallest positive integer $n$ such that $ng \in H$ for all $g \in G$?

Let $G$ be an abelian group and $H$ a subgroup. What is the smallest positive integer $n$ such that $ng \in H$ for all $g \in G$? Is it $[G : H]$, or can it be strictly smaller (a divisor of $[G : ...
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1answer
233 views

Find the Glide Line and Glide Vector

This is a question from a homework assignment. I have been spinning my wheels for some time and could really use a hint. Here is the set up. Let $t_a$ denote the isometry of translation by a vector ...
2
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0answers
32 views

Infinite fields of characteristic $p$ [duplicate]

I need to find two infinite fields of characteristic $p$ which are not isomorphic.
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1answer
145 views

Flexes of cubic curve

Which are the flexes of the cubic curve of Fermat $$x^3+y^3+z^3=0$$ at $\mathbb{P}^2(\mathbb{C})$ ? Could you give me a hint how we could find the flexes? Do we have to use maybe the following ...
2
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2answers
206 views

Order of general- and special linear groups over finite fields.

Let $\mathbb{F}_3$ be the field with three elements. Let $n\geq 1$. How many elements do the following groups have? $\text{GL}_n(\mathbb{F}_3)$ $\text{SL}_n(\mathbb{F}_3)$ Here GL is the general ...
3
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1answer
32 views

Order of $\mathrm{GL}_n(\mathbb F_p)$ for $p$ prime [duplicate]

While proving some facts about matrix group operations on finite fields, I stumbled across the following question: What is the order of the group of invertible $n\times n$ matrices over a ...
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1answer
39 views

NTRU cryptosystem

In the NTRU cryptosystem we are dealing with convolution polynomial rings and we compute $f(x)= T(d+1,d)$ and $g(x)= T(d,d)$ but when calculating their inverse in $R_q=(Z/qZ[x] / (x^N-1))$ and ...
0
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2answers
46 views

$S$ be the collection of groups $G$ in which every element in $G$ commutes only with the identity element and itself [closed]

Let $S$ be the collection of (isomorphism class of ) groups $G$ which have the property that every element in $G$ commutes only with the identity element and itself. Then A. |S|=$1$ ...
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0answers
35 views

Algebraic Structures Books [duplicate]

I wanted to ask you guys if you know any books where I can learn basic stuff about Algebraic Structures and Groups. Thank you.
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2answers
52 views

Show that $H$ is not normal in $A_4$?

Let $H=\{(1), (1,2)(3,4)\}$ in $A_4$. Show $H$ is not Normal in $A_4$ using the definition of normal. So I know that $A_4$ is the alternating group on $4$ letters, but I don't understand what that ...
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1answer
33 views

Commutative ring is sum of two ideals iff $x \to (x + I, x + J)$ is surjective.

I'm stuck on this exercise and any help would be well appreciated: Let $R$ be a commutative ring with ideals $I,J$. Show that $R=I+J$ if and only if $\phi(x)= (x + I, x + J)$ is surjective from ...
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2answers
29 views

S3 group action faithful?

I'm struggling with understanding the term "faithful". I read that a group action for example $S_3$ is faithful on {1,2,3}. Does that mean $S_3$ is not faithful on {1,2,3,4} because it never changes ...
3
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3answers
140 views

Argument for subgroup of a group

For a fixed element $a$ of a group $G$, prove or disprove that the set $H =\{xa|x \in G\}$ is a subgroup of $G$. So one argument I proved it using regular properties, but I was thinking in this ...
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2answers
38 views

let $G$ to be finite abelian group of order O(G), let n to be prime number and (O(G),n)=1 prove that $g=x^n$ for any $ x \in G$

let $G$ to be finite abelian group of order O(G), let n to be prime number and (O(G),n)=1 prove that $\forall g \in G$ we can write $g=x^n$ for any $ x \in G$
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4answers
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To which group is $G/\ker \phi$ isomorphic to?

$G = \langle H, \odot_7\rangle$ where $H= \{ 1,2,3,4,5,6\}$ and $\odot_7$ denotes the operation, multiplication modulo $7$, the function $\phi: G \to G$ defined by $\phi(g)=g^2$ . List the elements ...
0
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2answers
33 views

Tower of fields

Let $\mathbb{F}$ be a field and $\mathbb{K}$ be its extension and $\alpha$ $\in$ $\mathbb{K}$. Now, we know that $\mathbb{F}$ $\subset$ $\mathbb{F(\alpha)}$. But why $\mathbb{F(\alpha)}$ $\subset$ ...
5
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1answer
365 views

Prove $G/\ker \phi \times \ker \phi \cong G$

If $G, H$ are groups and $\phi : G \to H$ is a homomorphism, is it true that $G/\ker \phi \times \ker \phi \cong G$? I am pretty sure this is right, but I can't remember how to prove it. We can ...
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0answers
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Abstract algebra about module [closed]

let M be a left Q-module.show that the given action of Q is the only one which can be used to make M a left Q-module. I can prove z-module ,z stands for integer .
2
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3answers
39 views

a question about abstract algebra, prove that $HK\cong H\times K$

Let $H$ and $K$ be subgroups of a group $G$, $HK=KH$ and $H\cap K=\{1\}$. Prove that $$HK\cong H\times K.$$ Can some one tell me how to prove this question? I have spent too much time in it, ...
4
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1answer
335 views

The UFD field lemma

This page contains a result which it refers to as the UFD field lemma. I was wondering if anybody knew of any other references which discuss this result--this page is the only place I've seen it. The ...
3
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1answer
51 views

Polynomial Pell equation

Can someone point me in the right direction? Let $k$ be a field of characteristic $0$ and let $D \in k[x]$ be non-constant. Prove that the ‘polynomial Pell’ equation $$f^2 − Dg^2 = 1,\,\,\,\,f, ...
2
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0answers
48 views

$2$-groups with odd permutations

If $P$ is the Sylow $2$-subgroup of a finite group $G$, $H <P$, and $x \in P$ so that no non-trivial element of $\langle x \rangle$ conjugates into $H$ (in $G$), and $|P|=|H||x|$, how can I show ...
6
votes
1answer
141 views

Abelian subgroup in a 2 group.

Let $G$ be a non-abelian 2-group of order greater than or equal to 32 and $|Z(G)|=4$. Does the group $G$ has an abelian subgroup $H$, such that $16 \leq |H| \leq |G|/2$?
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1answer
66 views

on automorphisms groups a finite 2-group

Let $G=\langle a,b\mid a^4=b^4=1, bab^{-1}=a^3\rangle$. Please prove that $Aut(G)$ is generated by the automorphisms $$a\mapsto ab,\hspace{10pt} a\mapsto a^3,\hspace{10pt} a\mapsto ab^2, ...
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0answers
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How do I naively compute Gröbner bases?

I have a upcoming test tomorrow. In my class, my professor didn't give us complete proofs but taught us how to compute Gröbner base and he told me computing problems are gonna be on exam. I hate this ...
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1answer
41 views

Gaussian for Grassmann variables

Let $(\theta,A\theta)=\theta_i A_{ij}\theta_j$ where $A$ is some $(2\times2)$ antisymmetric matrix. I want to generalize the following $$I(A) =\int d\theta_1d\theta_2~ ...
2
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1answer
48 views

Find the characteristic of $Z_n \times Z_m$:

so I was given the problem: find the characteristic of $Z_3\times Z_4$ and I got $\operatorname{char}(Z_3\times Z_4)=12$, is it true that for any $Z_n \times Z_m$, $\operatorname{char}(Z_n \times ...
5
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2answers
481 views

Does localization preserve reducedness?

Is the localization of a reduced ring (no nilpotents) still reduced?
2
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1answer
286 views

Abstract Algebra - Cyclic Subgroups with a Proper Subgroup of Infinite Order

If a cyclic subgroup has a proper subgroup of infinite order, how many finite subgroups does it have? Explain. Any help appreciated! Thank you.
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0answers
21 views

Homology ring and cohomology ring

If I have that the cohomology ring of something has $0$ in all odd dimensions and $\mathbb{Z}$ in all even dimensions, and the the ring is isomorphic to $\mathbb{Z}[x_1, x_2, ..., x_i, ...]$, is it ...
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2answers
69 views

Are polynomials over $\mathbb{R}$ solvable by radicals?

I know that if $f$ is a polynomial over a subfield $F$ of $\mathbb{C}$ and $f$ is solvable by radicals, then the Galois group of $f$ over $F$ is solvable. I've also seen many applications of this fact ...
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0answers
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Show that the complex representations of a finite abelian group G are closed under pointwise multiplication [closed]

Show that the complex representations of a finite abelian group G are closed under pointwise multiplication, and form a group isomorphic to G. Show that representations of G are self-dual iff every ...
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1answer
54 views

Odd order n smaller than 27

I have a group $G$ that is a group of matrices of the form $$\left( \begin{array}{ccc} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{array} \right)$$ where $a,b,c \in \Bbb Z_3$. ...
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1answer
43 views

Show that $E=\mathbb{Q}(a)$

Let $f(x) \in \mathbb{Q}[x]$ an irreducible polynomial with the splitting field $E$ and let the group $Gal(E/\mathbb{Q})$ be abelian. If $a$ is a root of $f(x)$ then $E=\mathbb{Q}(a)$. Could you ...