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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Is there a criterion for such black/white stone game?

Black and white stones arranged as $m$ row and $n$ columns. At each move, you could choose either one row or one column, and reverse each stone's color -- turn white stones to black, and black ...
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24 views

Let $S$ be a subset of a group $G$ that contains the identity element $1$ and such that the left cosets $aS$ with $a$ in $G$, partition $G$.

Let $S$ be a subset of a group $G$ that contains the identity element $1$ and such that the left cosets $aS$ with $a$ in $G$, partition $G$.Prove that $S$ a is a subgroup of $G$. My try: For $h$ in ...
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Proving that either $x $ or $2x$ is a generator of the cyclic group $(\mathbb Z_3[x]/\langle f(x) \rangle)^*$

if $f(x)$ is a cubic irreducible polynomial over $\mathbb Z_3$, prove that either $x $ or $2x$ is a generator of the cyclic group $(\mathbb Z_3[x]/\langle f(x) \rangle)^*$ Attempt: $f(x) = \alpha ...
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38 views

Reference for book on fundamental abstract algebra topics

Can anybody suggest a good book on the topics listed below? A single book would be preferable. Thanks. Groups, subgroups, normal subgroups,cosets,Lagrange’s theorem, rings and their properties, ...
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27 views

Looking for a alternative forms of p = x(x+p)^y [on hold]

Are there some "useful", equivalent forms of $p = o(o+p)^i$? A closed form (i.e. $f(p) = g(i, o)$) would be awesome, but any equivalent forms might also be valuable. This equation arises from an ...
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12 views

Proving something is a matroid

I am taking a matroid theory class, and I am having trouble understanding an example we did in class: Let $F$ be a field, $E$ a ground set, and $V$ a vector space over $F$. Let $\phi : E ...
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38 views

Is there a theory of “derived extensions”?

Given an exact sequence of groups $$1\rightarrow N\rightarrow G\rightarrow K\rightarrow 1$$ we call $G$ a central extension of $K$ by $N$ if the image of $N$ is contained in the center of $G$. Central ...
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20 views

What kind of structures can I count using Alg?

I'm interested in counting structures that satisfy certain constraints up to isomorphism. For example, I might want to know how many clutters there are on $n$ vertices. The only way I can think to ...
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22 views

Presence of non square elements in $GF(p)$

I came across a problem which had a line of explanation as follows : Let $a \in GF(p). a$ is a non square in $GF(p),~ p \neq 2 \implies \nexists~ b\in GF(p)~~|~~a =b^2 $. But, is it really possible ...
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34 views

$H$ not closed under addition due to inverses, but closed under inverses

I have a fairly basic question. Problem from my text: $G=\left \langle \mathbb{R}^2 ,+\right\rangle, H=\{(x,y):x^2+y^2>0\}. $ Determine whether H is a subgroup of G. It's easy to show that ...
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41 views

For a discrete valuation ring to be a PID, must it have an element of valuation 1?

When is a discrete valuation ring a PID? Must it have an element of valuation 1 or is this not necessary?
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44 views

Every irreducible polynomial of degree $m$ over $\mathbb F_p$ divides $x^{p^m}-x$

We consider $F=\mathbb F_p$ for $p$ prime, $f(x)$ an irreducible polynomial of degree $m$ over $F$ and $g(x)=x^{p^m}-x$. I want to show that $f(x)\mid g(x)$. From the fact that the field ...
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1answer
18 views

Bass numbers of a minimax module

Let $R$ be a commutative Noetherian ring, and $M$ be a minimax $R$-module. Are the Bass numbers of $M$ are finite? (An $R$-module $M$ is called minimax, if there is a finite submodule $N$ of $M$, such ...
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17 views

A non-$\text{UFD}$ where there exists a set $X$ of infinite cardinality of elements such that $a^2 | b^2$ does not lead to $a|b$ [duplicate]

Is there a commutative non-$\text{UFD}$ ring such that there exists a set $X$ of infinite cardinality of elements that for $\forall x \in X$, $x^2$ is a multiple of $a^2$ for some particular $a$, but ...
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36 views

Modules with finite support in $\mathrm{Max}(R)$

Let $R$ be a commutative Noetherian ring, and $M$ be an $R$-module. Is the following statement true? If $\mathrm{Supp}_R(M)$ (support of $M$) is a finite subset of $\mathrm{Max}(R)$ (the set of all ...
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55 views

If $x^2 a x=a^{-1}$, then $a$ has a cube root. [duplicate]

In a group $G$: If $x^2 a x=a^{-1}$, then $a$ has a cube root. (Hint: Show that $xax$ is a cube root of $a^{-1}$.) So essentially $\exists y\in G:a=y^3$. The hint probably confused me more than ...
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If $a^3=e$, then $a$ has a square root.

Assuming $a\in G$ where $G$ is a group. I'm not sure why this is hard for me. Essentially, the problem is just saying: If $a^3=e$, then $\exists x \in G : a=x^2$. Can somebody give me a hint or a ...
0
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1answer
57 views

Is it true that a field is a vector space over a field? [duplicate]

Is it true that a field is a vector space over a field? This idea arises in me after reading the solution for the question the order of finite field is $p^n$. Order of finite fields is $p^n$ I am ...
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1answer
35 views

Finding complements of direct summands

Let $B=\mathbb Z⊕\mathbb Z_4$. How could we prove that $B_1=(1,\bar 1)\mathbb Z$ and $B_2=(1,\bar 2)\mathbb Z$ are direct summands in $B$? Or, the same question for $A=\mathbb Z⊕\mathbb Z$ and ...
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Construction of the simply connected Lie group of a given Lie algebra

Given a finite dimensional real Lie algebra $\mathfrak{g}$, I am trying to obtain a concrete realization of its simply connected Lie group $G$, with $\mathrm{Lie}(G) \cong \mathfrak{g}$. Let us ...
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1answer
26 views

Example of commutative algebra over integers where there exists $x$ such that $x = y^2$ for several $y$'s

Is there a commutative algebra over integers such that there exists $x$ with $x = y^2$ for several $y$'s? Also, is there a commutative algebra over integers such that for every $k \in \mathbb{N}$, ...
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2answers
51 views

Is Hom$(G_1, G_2)$ a group?

The collection of all homomorphisms from the group $G_1$ to the group $G_2$ is denoted as Hom$(G_1, G_2)$. I am willing to show that if $G_1 \simeq G_1'$ then Hom$(G_1, G_2) \simeq$ Hom$(G1', G2)$. ...
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A central division algebra is not its commutator

In looking at old qualifying exam questions, I've come upon a question that has me stumped. Let $A$ be a central division algebra (of finite dimension) over a field $k$. Let $[A,A]$ be the ...
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1answer
63 views

Are there at least denumerably many distinct group operations on any denumerable set?

I'm working on a proof of the following statement: For any denumerable set $D$, there exist at least denumerably many distinct group operations on $D$. My argument is looking fairly messy, so I'm ...
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1answer
46 views

Why this order is well-defined?

I'm trying to understand why this order defined in Fulton's algebraic curves book is indeed well-defined: I have two questions: Why $u\in \mathcal{O_P}(X)$ (he is implicitly assuming this fact so ...
2
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3answers
57 views

Irreduciblility of $x^3 + 9x + 6 $ in $\mathbb{Q}[x]$

I am trying to prove the irreducibility of $x^3 + 9x + 6 $ in $\mathbb{Q}[x]$ without using Eisenstein's criterion. What I have done is -- Let assume it is reducible in $\mathbb{Q}[x]$, then it can ...
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$(a_1 a_2 \cdots a_n)^2=e$ in a finite abelian group

Let $G$ be a finite abelian group. Prove that $(a_1 a_2 \cdots a_n)^2=e$. My proof: $$\forall a \in \{a_1,a_2,\cdots,a_n\} \exists ! a^{-1} \in \{a_1,a_2,\cdots,a_n\}:a^{-1}a=aa^{-1}=e$$ hence, ...
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3answers
58 views

Group of even order must contain $a:a=a^{-1}$ $ (a\not = e)$ [duplicate]

Let $G$ be a finite group. If the order of $G$ is even, prove that there is at least one element $a$ in $G$ such that $a\not= e$ and $a=a^{-1}$. Here's my idea: Suppose $\{x_1,\cdots,x_n\}$ is ...
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1answer
33 views

Relating the characteristic of the ring R to the characteristic of R[x]

Suppose $R$ is a ring and $R[x]$ is the ring of polynomials in the indeterminate $x$ with coefficients from $R$. The characteristic of a ring is the smallest positive integer $n$ such that $n \cdot r ...
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Construction of Hyper-Complex Numbers

How does one construct a hyper-complex number multiplication table? For example: Quarternions: ...
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32 views

Exercise on localization as a colimit

I am doing the following exercise: Suppose $S$ is a multiplicative set of $A$, an integral domain, and interpret $S^{-1}A = \varinjlim \dfrac{1}{s}A$, where the limit is over $s \in S$ and in the ...
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Resultants of two polynomials over a ring

Let $k$ be a field $f,g\in k[x,y]$ be two polynomials. The resultant $R\in k[x]$ is a polynomial function of the coefficients of $f$ and $g$, such that $f$ and $g$ gave a common zero (in an extension) ...
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1answer
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Notation for vector space of polynomials of bounded degree

Is there standard notation for the vector space of polynomials in $n$ variables with coefficients in a field $F$ and with degree at most $D$? Without bounding the degree, it is $F[x_1, \ldots, x_n]$. ...
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68 views

An epimorphism from $\mathbb Z⊕\mathbb Z⊕\cdots$ to $\mathbb Q$

I want an explicit example of an epimorphism from $\mathbb Z⊕\mathbb Z⊕\cdots$ to $\mathbb Q$. Thanks.
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Rapid and easy question on ideals and ring

Let $R$ be the number ring related to a field $K$ of finite degree over $\mathbb Q$, i.e. $\mathbb Q\le K\le\mathbb C$ and $[K:\mathbb Q]=n$. Hence $R=\mathbb A\cap K$, where $\mathbb A$ is the ring ...
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1answer
21 views

Rational group algebras and maximal orders

Let $G$ be a finite group, and $\mathbb{Q}[G]$ be the rational group algebra. Then the group ring $\mathbb{Z}[G]$ is an order in $\mathbb{Q}[G]$, but is not in general a maximal order. What can we ...
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Non-UFD that there exists set $X$ of any cardinality and $a$ that $xy$ is not divisible by $a^2$ for any $x,y \in X$ and $x^2$ is divisible by $a^2$

Let's say we want to construct a non-UFD that is a commutative ring that satisfies: There exists $a$ such that there exists a set $X$ of elements of finite cardinality $k$ such that for any $x,y \in ...
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1answer
32 views

One-sided and two-sided cancellability

Is there a semigroup $S$ with right- and left-cancellable elements but no elements cancellable from both sides? $s\in S$ is left-cancellable if for any $a,b\in S$ we have $$sa=sb\implies a=b$$ and ...
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1answer
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Injective hull of $\mathbb{ Z}_n$ [duplicate]

What is the injective hull of $\mathbb Z_n$? I know that in case $n=p$ is prime, the injective hull would be isomorphic to $\mathbb Z_{p^∞}$, but in general case, I have no idea. Can anyone be of ...
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An estimate involving gaps in a subsemigroup of $(\mathbb N,+)$

I think this question can be solved by a high school student, maybe there is some trick on it or I'm forgetting something. Before my question, some background is required: Definition: A ...
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Converting a polynomial ring to a numerical ring

One may be curious why one wishes to convert a polynomial ring to a numerical ring. But as one of the most natural number system is integers, and many properties of rings can be easily understood in ...
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moufang loops and invertible real octonioms

Let Q be a Moufang loop of invertible real octonions. Which variety of Moufang loops is generated by Q? All of Moufang loops?
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1answer
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Isomorphism between $R$ and its dual space

Let $R$ be a finite dimensional algebra over a field $K$. If $f$ is an $R$-module monomorphism from $R$ to the dual $K$-space $\operatorname{Hom}_K(R,K)$ why it is onto? Thanks!
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2answers
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A non-UFD such that $a^2 \mid b^2$ does not lead to $a\mid b$

Is there any non-UFD that is a commutative ring such that $a^2 \mid b^2$ does not always lead to $a\mid b$? It would be preferable if examples are something that does not involve ...
4
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1answer
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Prove $g^2 = e$ if there is a subgroup of index 2 that does not contain $g$ for every $g \in G$.

I'm having some trouble with this question from a practice exam. Let $G$ be finite group. Suppose for every $g \in G$ other than the identity element $e$, there is a subgroup $H \subset G$ of index ...
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1answer
78 views

How does $A_n$ look in Aut$(X)$?

Let me phrase my question precisely: Let $X=\{1,2,3,...,n\}$, $ \ S_n=\mbox{Sym}\{1,2,3,...,n\}$ be symmetric group on $n-$letters. Let $\mbox{Aut}(X)$ denote the automorphism group of $X$. We ...
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Showing the existence of sub fields of a finite field

For each divisor $m$ of $n$, $GF(p^n)$ has a unique sub field of order $p^m$ . The proof of this theorem in Gallian goes like this : Suppose that $m$ divides $n$. Then, since : $(p^n-1) = ...
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homology commutes with direct sum and product?

I'm looking at exercise 1.2.1 from Weibel's Intro to Homological Algebra. (I need to show that homology commutes with direct sum and direct product.) Is it possible to show that cokernels commute with ...
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1answer
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Proving that the symmetric difference of sets is a group

I wanted to ask about this problem. My book states: Let the symmetric difference be defined as $A + B=(A\setminus B) \cup (B \setminus A)$. It proceeds to define a power set as $P_D= ...
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Typo in Marcus book “number fields”.

Page 133, six line from the bottom. He wrote $n=[L:K]$, but I think it's $n=[K:\mathbb Q]$. Am I right?