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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Category with only one object.

Suppose we have the category with only one object - the group G. Why can we think of the morphisms in this category as of the elements of the group G? I would be very grateful for explanation.
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A question on cyclic group with finite order

I have trouble proving the following statement: Suppose that $H$ is a finite group with order $n$ and $e$ is the identity element of $H$. For an arbitrary positive integer $d$ satisfing ...
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$\dim B/A=\dim B-\dim A$?

If $A,B$ are two vector spaces over $k$ such that $B\subseteq A$, can I say $\dim B/A=\dim B-\dim A$? I need of this result to prove a theorem I'm working on. Thanks in advance
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Groups with cyclic Commutator subgroup

Is anything known about class of groups with cyclic commutator subgroup?
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Kernel of a homomorphism: why $g_i(\alpha)\in Q_i$?

Let $K\le L$ be two number fields, $[L:K]=n$. Let $R=\mathbb A\cap K$ and $S=\mathbb A\cap L$ be the relative number rings. Take $\alpha\in S$ an element of degree $n$, i.e. such that $L=K[\alpha]$. ...
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Injectivity of Natural Homomorphism to Groupification

This is a continuation of my own question some time ago. Suppose $M$ is a monoid and $G$ is the groupification of $M$. (I figure groupification of $M$ is a better term than Grothendieck group of ...
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33 views

If field has a prime field isomorphic to $\mathbb{Q}$, sufficient condition for every subring being integrally closed domain

Suppose that a field $k$ has the prime field isomorphic to the field of rational numbers $\mathbb{Q}$. Then what would be sufficient condition in order for every subring of $k$ be integrally closed ...
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Isomorphism of Group Products

Let $G$ be a group, $A = G \times G$. In $A$, Let $T = \{(g, g)|g \in G\}$. Prove that $T$ is isomorphic to $G$. I don't know how to continue this problem. $A$ is abelian. Therefore, $G \times G$ is ...
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26 views

Field of prime characteristic over two indeterminates

Let $F$ have prime characteristic $p$ and let $E = F(Y,Z)$, where $Y, Z$ are indeterminates. Let $L=F(Y^{p} , Z^{p})$ $\subseteq E$. a. Show that $\alpha^{p} \in L$ for all $\alpha \in E$. b. Show ...
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31 views

When is a subring of a field an integrally closed domain? [on hold]

What criteria would be necessary/sufficient for a subring of a field to be an integrally closed domain?
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44 views

Condition on a field that makes every subring of the field an integrally closed domain?

Every subring of a field is an integral domain. Now I want to know what would need to be additionally imposed on the definition of a field to make every subring of the field an integrally closed ...
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86 views

In which Fields, does $x^n-x$ have a multiple zero?

In which Fields, does $x^n-x$ have a multiple zero? Attempt: Let $f(x) = x^n-x$ and $f'(x) = nx^{n-1}-1$ If $f(x)$ has a multiple zero, then, $f(x)$ and $f'(x)$ have a common factor. An irreducible ...
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finite group homology: $nH_k(G;M)=0$ for $n=|G|$?

Let $G$ be a finite group. Is there a simple proof (if any) that the order of $G$ annihilates the Eilenberg-MacLane homology $H_k(G;M)$ for all $k\geq1$? A simple proof of the statement for ...
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1answer
31 views

Is it possible to represent subsets of natural numbers as groups with prime generators?

I'm learning group theory and I'm trying to consider the "symmetry" of a certain group of natural numbers: Here's the idea, all natural numbers are comprised of multiples of primes. So a subset would ...
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Exercise on characterization of free abelian groups

I was wondering if someone can please check my work on a homework problem. This is from the graduate Hungerford text. Chapter 2.1, number 3. Let $X=\{a_i\ |\ i\in I\}$ be a set. Then the free abelian ...
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prove that either $f=I_{\mathbb Z}$ is the identity function or $f(x)=0, \forall x \in \mathbb Z$.

Let $f:\mathbb Z \rightarrow \mathbb Z$ such that $f(x+y)=f(x)+f(y), \forall x,y \in \mathbb Z$ and $f(xy)=f(x)f(y), \forall x,y \in \mathbb Z$. I need to prove that either $f=I_{\mathbb Z}$ is the ...
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What would be a prime element in the field of rational numbers?

It is clear to understand what prime elements will be in case of ring of integers and many other rings. However, I find it confusing in case of fields, more specifically field of rational numbers. So ...
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Regular function on a variety which is not globally rational

I am looking for a particularly simple example of a regular function $f : V \to \mathbb{A}^1_k$ for some affine variety $V \subseteq \mathbb{A}^n_k$ over a field $k$, which cannot be expressed by a ...
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1answer
14 views

A hyperplane in a $k$-algebra

Let there exist a nonsingular bilinear pairing $B:R×R→k$, where $R$ is a finite dimensional algebra over a field $k$, such that $B(xy,z)=B(x,yz)$ for all $x,y,z$ in $R$. Why the set $\{z∈R∶B(1,z)=0\}$ ...
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45 views

Tensor product of Frobenius algebras

In proving the fact that the tensor product of any two finite-dimensional Frobenius algebras $R$ and $S$ over the same field $k$, it is usually defined a $k$-bilinear pairing $E: W×W→k$ where ...
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Extending a field homomorphism to a polynomial ring [on hold]

please help me do this Let $f : k \to K$ be a homomorphism of fields, and let $a\in K$. Prove that there exists a unique homomorphism $g: k[x] \to K$ such that $g_{|k} = f$ and $g(x) = a$.
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Equality of subgroups $H \subseteq K \subseteq G$ with the same finite index in $G$

Let $G$ be a group and let $H,K$ be two subgroups of $G$ such that $H\subseteq K$and $[G:H] = [G:K]$ is finite. Prove that $H = K$. Can somebody please give me some idea to solve this?
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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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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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38 views

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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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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35 views

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

Are there some "useful", equivalent forms of $p(i,o) = o(o+p(i,o))^i$? A closed form (i.e. $p(i, o) = g(i, o)$) would be awesome, but any equivalent forms might also be valuable. This equation ...
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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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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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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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28 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 $. But, is it really possible ...
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$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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51 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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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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Bass numbers of minimax modules are finite?

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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A non-UFD where there exist infinitely many elements such that $a^2 \mid b^2$ does not lead to $a\mid b$ [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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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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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 ...
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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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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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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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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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66 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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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 ...
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3answers
61 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 ...