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

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Is there a redundant assumption in Exercise 2, page 14, from Janusz “Algebraic Number Fields”?

The exercise says: Let $R$ be an integral domain with quotient field $K$ and let $M$ be an $R$-submodule of a finite dimensional $K$-vector space. Prove $M=\bigcap_{P} R_P M$, where the ...
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1answer
38 views

A solvable quintic with the root $x=(\sqrt[5]{p}+\sqrt[5]{q})^5$ - what are the other roots?

I derived a two parameter quintic equation with the root: $$x=(\sqrt[5]{p}+\sqrt[5]{q})^5,~~~~~p,q \in \mathbb{Q}$$ $$\color{blue}{x^5}-5(p+q)\color{blue}{x^4}+5(2p^2-121pq+2q^2)\color{blue}{x^3}...
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1answer
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Empty set in an algebra or sigma-algebra

Does an algebra (or a sigma-algebra) contains the empty set or a set containing the empty set? E.g., let $X$ be a set. Is the trivial sigma-algebra $\{\emptyset,X\}$ or $\{\{\emptyset\},\{X\}\}$?
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Workability of linear equation solving methods for different fields?

So far, I have mostly done linear algebra over $\mathbb R$ and $\mathbb{C}$. I know there exist very many methods to solve equation systems for those fields, of which a few are Gaussian Elimination, ...
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27 views

Sylow subgroup of a symmetric group

Consider the symmetric group of$S_{20}$ and it's subgroup $A_{20}$ consisting of all even permutations. Let $H$ be a $7$-Sylow subgroup of$A_{20}$. Is $H$ cyclic? And is correct the statement which ...
2
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1answer
28 views

Not getting how to prove reverse hypothesis.

This is a theorem from Dummit & Foote text- Let $G$ be a group acting on the non-empty set $A$.The relation on $A$ defined by $a \sim b$ iff $a=g.b$ for some $g \in G$ is an ...
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0answers
33 views

What is the algebraic structure of $\Bbb Q_p/\Bbb Z_p$? [on hold]

I am curious about the algebraic structure of $\Bbb Q_p/\Bbb Z_p$. Is there any result in this direction? Thanks!
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1answer
104 views

Only five solvable quintic equations of the form $x^5+ax^2+b=0$? What are their solutions?

According to Wikipedia there is only five solvable quintic equations of the form $x^5+ax^2+b=0,~~a,b \in \mathbb{Q}$ (up to a scaling constant $s$). $$x^5-2s^3x^2-\frac{s^5}{5}=0 $$ $$ x^5-100s^3x^2-...
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0answers
36 views

Simmetric group exercise of an exam [on hold]

In $S_5$ let be $\sigma=(12435)$ and $\tau=(25)(34)$, and $H= <\sigma, \tau>$. Show that $N(<\sigma>) = H$ N.B. $N(<\sigma>)$ is normalizer in $H$, not in S5. Deduce that $H=&...
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0answers
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Can i say that ( U(Zm) , * ) is isomorphic to ( Zk , +)

Can i say that ( U(Zm) , * ) is isomorphic to ( Zk , +) where k=phi(m) or to Za x Zb x Zc x....where abc..=k with the right combination of a,b,c... ?
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1answer
31 views

How to find a modulus equation? [on hold]

Let $x$ and $q$ be an integer.Also, $a$ and $b$ are integers. We know the two modulus equations. i) $x \equiv y$ mod $p^a -1$ ii) $x \equiv z$ mod $p^b -1$ Then how to find $x$? Can we find $x$ ...
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2answers
63 views

Where to learn the algebra behind the use of differential operators in calculus

Algebra with differential operators is often used as a shortcut in calculus problems. I have previously asked about manipulations such as these: $$\int x^5e^xdx =\frac1Dx^5e^x=e^x\frac{1}{1+D}x^5=e^x(...
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0answers
23 views

Symmetric brace algebras - unshuffle sequences

I'm studying brace algebras in this article: Symmetric Brace Algebras. In the following definition, what do the authors mean by "unshuffle sequences"? Definition 2. A symmetric brace algebra is a ...
3
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2answers
98 views

When a prime ideal in polynomial ring over integers is principal [duplicate]

While dealing with a question about a prime ideal $I\subset\mathbb{Z}[x]$ (with $0$ in $I$ as the only constant polynomial) I was asked to show that there exist $f(x)\in\mathbb{Z}[x]$ such that $I=\...
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1answer
29 views

quick question on ascending chain condition for rings

I know that if $R$ is a commutative ring with an identity in which every ideal if finitely generated then it satisfies the ascending chain condition. Just wondering if the converse is also true?
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Question on inverse limits

1.7. Remark. The inverse limit of an inverse system of non-empty sets might be empty as the following example shows: Let $I:=\mathbb{N}$ and $X_n:=\mathbb{N}$ for every $n\in\mathbb{N}$. Let $\...
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3answers
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Every subspace is the kernel of a linear map

I know that every kernel of a linear map from $\mathbb R^n$ to $\mathbb R^m$ is a subspace of $\mathbb R^n$. I am wondering if the converse is true, i.e. every subspace of $\mathbb R^n$ is the kernel ...
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1answer
24 views

If a normal series has maximal lengh in $G$, then it is a composition series.

A normal series of subgroups of $G$ is a decreasing sequence of subgroups $... \subset G_1 \subset G_0 = G$ where $\subset$ denoted strict inclusion and $\forall i \ \ G_i$ is normal in $G_{...
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2answers
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Automorphism of Upper half plane

Let $M=\left\{\left.\displaystyle z\mapsto\frac{az+b}{cz+d}\ \right|\ \ ad-bc\not =0\right\}$,$$p:GL(2,\mathbb C)\to M, \begin{bmatrix}a & b \\ c & d \end{bmatrix}\mapsto\frac{az+b}{cz+d}.$$ ...
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45 views

Finding nilpotent elements in a quotient ring.

Which are nilpotent elements of $\mathbb{Q}[x]/(x^5-3x^2)\times\mathbb{Z}/(12)$? I tried to decompose in this way: $$\mathbb{Q}[x]/(x^5-3x^2)\times\mathbb{Z}/(12)\cong\mathbb{Q}[x]/(x^2)\times\mathbb{...
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A field in which every element (that is not 1 or 0) is a root of -1

Let $\mathbb{F}$ be a field with $char(\mathbb{F}) \neq 2$ such that for every element $q \in \mathbb{F}$ if $q \neq 0$ and $q \neq 1$ then there is a power n such that $q^n = -1$. (E.g. $\mathbb{F}_3$...
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1answer
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If $R$ is an integral domain with unity having only finitely many subdomains (not necessarily with unity), then is $R$ finite?

If $R$ is an integral domain with unity having only finitely many subdomains (not necessarily with unity), then is it true that $R$ is finite ? (I know that there are infinite domains with unity, ...
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3answers
52 views

$D$ be a UFD, if an element of $D$ is not a square in $D$ then is it true that, that element is not a square in the fraction field of $D$?

Let $D$ be a UFD, let $F$ be the field of fractions of $D$, let $a \in D$ be such that $x^2 \ne a, \forall x \in D$. Then is it true that $x^2\ne a ,\forall x \in F$ ? (This problem is motivated ...
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1answer
44 views

Modules over algebras vs Modules over Rings?

Since I've started module theory I was confused with a point. What is more general, the theory of modules over algebras or over arbitrary rings? I hope this is not a pointless question so let me try ...
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126 views

Is there a subject in mathematics like topological Algebra?

I would consider myself an algebraic topologist and there is a lot of influence from algebra into topology and without this input from the algebraic site I would say that a lot of topological theorems ...
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1answer
27 views

Need help for the example on conjugation.

This is an example from Dummit & Foote text, i've some queries in this- If $|G|>1$, then unlike action by left multiplication,$G$ does not act tranistively on itself by conjugation because {1}...
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Principal ideal of $\mathbb{C}[Z,\bar{Z}]$

Let $I$ be an ideal of $\mathbb{C}[Z,\bar{Z}]$. How to prove that $I$ is principal in $\mathbb{C}[Z,\bar{Z}]$ ? It exists some simple criterion to say that an ideal will be principal or not?
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Delooping of a ring?

I'm no expert on category theory, so the definition of delooping in the nlab article is a bit over my head. However, I do understand the practical idea that we can think of a group $G$ as a one-object ...
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99 views

Motivation behind word quotient

Why set of all cosets of a subspace W of a vector space V is called quotient space. What is the motivation behind word quotient?
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49 views

Minimal prime ideal and sum of two ideals [on hold]

Let $R$ be a commutative ring with $1$, and let $p$ be a minimal prime ideal of $R$. If $p\subseteq I_1+ I_2$, where $I_1$ and $I_2$ are two ideals of $R$, can we deduce that $ p\subseteq I_1 $ or $...
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1answer
48 views

A property of minimal prime ideals [on hold]

Let $R$ be a commutative ring with $1$, and let $\frak{p}$ be a minimal prime ideal of $R$. If $\mathfrak{p}\subseteq I_1\cap I_2$, where $I_1$ and $I_2$ are two ideals of $R$, can we deduce that $ \...
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1answer
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Show how (0,(12)) and (1,(12)) are in different conjugacy classes.

$S_3=${${(1),(12),(13),(23),(123),(132)}$}. $\mathbb Z_2=${${0,1}$}. $\mathbb Z_2$$\oplus$$S_3$={$(0,(1)),(0,(12)),(0,(13)),(0,(23)),(0,(123)),(0,(132)),(1,(1)),(1,(12)),(1,(13)),(1,(23)),(1,(123)),(...
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$\mathbb{Z}$ has no composition series. Need an assistance in some questions.

Please, read the whole post before trying to answer. Remark: here $\subset$ means "strict inclusion". I need to prove that the group $\mathbb{Z}$ has no composition series. That is, no normal series ...
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40 views

Class-equation of $\mathbb Z_2$ $\oplus$ $S_3$.

$S_3=${${(1),(12),(13),(23),(123),(132)}$}. $\mathbb Z_2=${${0,1}$}. $\mathbb Z_2$$\oplus$$S_3$={$(0,(1)),(0,(12)),(0,(13)),(0,(23)),(0,(123)),(0,(132)),(1,(1)),(1,(12)),(1,(13)),(1,(23)),(1,(123)),(...
2
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1answer
34 views

Isomorphic quotient rings of polynomial rings

I wish to know if $$\dfrac{\mathbb Z_2[x_1,x_2,x_3,x_4]}{\langle x_1x_3,x_2x_4,x_1+x_3,x_2+x_3+x_4\rangle}\cong\dfrac{\mathbb Z_2[a,b]}{\langle a^3,b^2,a^2-ab\rangle}.$$ Context - I was computing the ...
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5answers
275 views

Why are an even number of flips required to get back to the original list?

Consider the list of numbers $[1, \cdots, n]$ for some positive integer $n$. Two distinct elements $i$ and $j$ of the list can be switched in a so-called flip. For example, let $f$ be a flip that ...
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2answers
66 views

Let $R$ be a ring with unity. Consider $X^2 - 1$ $\in$ $R[X]$. Then $X^2 - 1$ has at most two roots in R.

Indicate True/False Let $R$ be a ring with unity. Consider $X^2 - 1$ $\in$ $R[X]$. Then $X^2 - 1$ has at most two roots in R. I need a hint to solve this problem. I have tried some common rings ...
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1answer
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Criterion of separability of polynomials

When I study page 270 in Lang's algebra, I have a problem. Let $f(X)=X^3+aX+b$ be an irreducible polynomial over a field $k$. Lang says that if char $k$ is not equal to $2,3$, then $f$ is separable. ...
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1answer
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What makes an algebra finite?

What makes an arbitrary algebra finite? If an algebra $A$ is generated by an infinite set of generators, but the operations of $A$ are finitary, is $A$ finite or infinite? If $A$ is generated by a ...
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38 views

On oblath's theorem [on hold]

it is just my first encounter about this topic ,it is the topic that my prof gave to me in my undergrad studies.I found it interesting but there are still parts(like theorem) in this topic which make ...
2
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2answers
42 views

Operation of permutations on functions

Let $P$ be the additive group of mappings from $\mathbf{Z}^n$ to $\mathbf{Z}$. For $f \in P$ and $\sigma \in \mathfrak{S}_n$ (the symmetric group of degree $n$) let $\sigma f$ be the element of $P$ ...
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2answers
26 views

A simple group such that $[G:H]=n$ can be embedded into $A_n$

Let $G$ be a finite simple group and $H$ be a proper subgroup of $G$ such that $|G:H|=n$. Then, how do I prove that $G$ can be embeded into $A_n$? I can prove that $G$ can be embedded into $S_n$ ...
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2answers
70 views

Polynomial ring with arbitrarily many variables in ZF

For a given field $k$ and a set $X$ we want to define the ring $k[X]$ of polynomials with $X$ as the set of variables. We do not assume $X$ to be finite. And we want to do this without employing axiom ...
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2answers
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Linking two theorems: on algebraic closure and on minimal splitting field

Consider following two statements from same book of Cohn: Basic Algebra. Proposition 7.3.2: If $k$ is any field and $\mathcal{F}$ is a set of polynomials over $k$, then any two minimal splitting ...
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21 views

Techniques to turn expressions involving integer roots into polynomials by substitution?

Inspired by this question involving an Equation for a Torus How to find a parametrization for a torus? I started wondering if there is some systematic approach to do substitutions to make equations ...
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64 views

Does associativity imply closure?

Does associativity of binary operation imply closure under this operation? Sometimes definitions of semigroup, group or vector space omit axiom of closure under corresponding operations and sometimes ...
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1answer
48 views

Deciphering the main theorem of the paper ''On Oblath's Problem''

I am trying to read the paper On Oblath's Problem, and I'm have difficulty understanding the main theorem. I can read the theorem but I don't understand it. May someone help me to make this theorem as ...
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2answers
30 views

Modules as morphisms to endomorphism rings

An $A$-module $M$ may be thought of as a (surjective) ring homomorphism $f: A \to E(M)$, where $E(M)$ is a ring of group endomorphisms of $M$. Then $am = f(a)(m)$. Is there any more to this ...
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50 views

There are 267 not isomorphic groups of order 64. How many of these are Abelian? Describe them. [closed]

There are 267 not isomorphic groups of order 64. How many of these are Abelian? Describe them.
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2answers
25 views

Group order 168 having a normal subgroup of order 4, then G has a normal subgroup of order 28.

Prove that if G is a group of order 168 that has a normal subgroup of order 4, then G has a normal subgroup of order 28. resolution: P4 is the ordeme subgroup 4 and P7 the order subgroup 7. As P4 is ...