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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Cubic field extension question

Is the following statement true ? $$ 2^{\frac{1}{3}} \in \mathbb{Q}(4^{\frac{1}{3}}).$$
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My answer to this simple question is wrong but I don't know why

I'm self-studying abstract algebra, and prior to fields there's a brief section on vector spaces. One of the questions asks: "Is $U = \{(a, b-1, c)| a, b, c \in F \}$ a subspace of $F^3$? ($F$ a ...
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abelianized fundamental groups.

I am trying to show that there is a canonical ismorphism between the abelianized fundamental groups, $\pi_1(X,p)_{ab}$ and $\pi_1(X,q)_{ab}$ of the path-connected space $X$. I know since $X$ is path ...
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If I define $ I.J=\{ij : i \in I $ & $ j \in J \} $. Then prove that it is not necessrily an ideal, where $I,J$ are ideals in a ring $R$. [duplicate]

If I define $ I.J=\{ij : i \in I $ & $ j \in J \} $. Then prove that it is not necessrily an ideal, where $I,J$ are ideals in a ring $R$. I have found one counter example in $R[x,y,z]$ for ...
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Showing $\operatorname{Aut}(\mathbb{R})$ is abelian

I have an exercise (not for a class) that asks whether $\operatorname{Aut}(\mathbb{R})$ (field automorphisms) is abelian. I know that $\operatorname{Aut}(\mathbb{R})$ is just the trivial group, but ...
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Direct product of nontrivial rings can never be an integral domain or a field [duplicate]

I saw this claim in Pinter's A book of Abstract Algebra. I cannot understand. If the two rings are both integral domains, the direct product should be an integral domain. If the two rings are both ...
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Normal Form of Elements in Quotient Groups

Let $G=⟨ S\mid R_1⟩$ be a group, where $S$ is the set of generators and $R_1$ is the set of relations. Let $H=⟨S\mid R_1, R_2⟩$ be the quotient group $G$ obtained from $G$ by adding a (possibly ...
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How do I prove that finitely generated group with $g^2=1$ is finite?

Let $G$ be a finitely generated group. Assume for all $g\in G, g^2=e$. Then, how do I show that $G$ is actually finite? I don't know where to start..
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How many group homomorphisms are there from Zn to Zm?

In looking up this question, I found this site: Physics Forums. In it, someone claims that $f(x) = kx$ is a homomorphism from the group $\mathbb{Z}_{m}$ to $\mathbb{Z}_{n}$ if $m$ divides $kn$. I ...
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whitney class of fiber bundles over classifying space of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$. Let $\rho: \Sigma_k\to GL(k)$ be regular representation by permuting basis. Let $\rho': B\Sigma_k\to BGL(k)=G_k(\mathbb{R}^\infty)$ be the induced ...
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938 views

A vector space is an abelian group with some extra structure?

Question: A vector space is an abelian group with some extra structure. Given two vector spaces V1 × V2, show that the group V1 × V2 is a vector space. Can someone explain to me the first sentence? ...
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Prove that: $A/<f.g>=A/<f>\times A/<g>$ with $f,g$ are coprime \. [on hold]

Prove that $A=\mathbb{Z}$ or $k[x]$ with $k$ is a field, and $f,g$ are coprime elements of $A$, then $A/<f.g>=A/<f>\times A/<g>$. With $<f>$ is an ideal.
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1answer
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An equivalence relation iff G≈H, where G and H are groups [duplicate]

Problem : Let $S$ be the relation G~H iff G is isomorphic to H. Show reflexive, transitivity and symmetric. First show G is automorphism, which will imply G~G. So the identity mapping gives us ...
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What is the tensor product of Z/10Z with Z/12Z? [duplicate]

I've just met tensor products of modules in part of my self-study and as a concrete exercise I've been asked to calculate $\mathbb{Z}/(10){\otimes}_{\mathbb{Z}} \mathbb{Z}/(12)$. Short of writing out ...
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Sum and product of comaximal ideals

Let $R$ be a commutative ring with unity. If $R=I_{i}+I_{j}$, for all $i\ne j$, where $I_1,I_2,...,I_n$ are ideals of $R$, I want to show that $$R=I_{n}+I_{1}I_{2}\cdots I_{n-1}.$$ I started off ...
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Find gcd and lcm of two polynomials

Let $f(x)=x^3+x^2+x+1$ and $g(x)=x^3+1$. Then in $\mathbb{Q}[x]$ $\gcd (f(x),g(x))=x+1$ $ \gcd(f(x),g(x))=x^3-1$ $\operatorname{lcm}(f(x),g(x))=x^5+x^3+x^2+1$ ...
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Quotient Field of an Integral Domain

The question is: Let $n$ be a positive integer and let $D$ be the subset of all rational numbers of the form $$\frac{a}{n^k}$$ with $a \in \mathbb{Z}$ and $k$ any positive integer. Show that $D$ ...
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Question about the cycle type of the Frobenius Automorphism

Let $f$ be an irreducible and separable polynomial of degree $n$ over $\mathbb{F}_p$. For a finite field $F$ of characteristic $p$, we define $\phi_p:\alpha\mapsto\alpha^p$. Then we know $\phi_p$ is ...
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$M$ is a simple module if and only if $M \cong R/I$ for some $I$ maximal ideal in $R$.

I am trying to show the following statement (taken from Rotman's Advanced Modern Algebra): Let $M$ be an $R$-module. Then $M$ is a simple module if and only if $M \cong R/I$ for some $I$ maximal ...
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order of a sylow p-subgroup

Let $p$ be a prime number . The order of a $p$-Sylow subgroup of the group $GL_{50}(F_p)$ of invertible $50\times50$ matrices with entries from the finite field $F_p$,equals which of the following ...
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Sylow's First Theorm in GRE book (typo?)

In my Gre book, the Sylow's First Theorem is stated as Let $G$ be a finite group of order $n$, and let $n= p^k m$, where $p$ is a prime that does not divide $m$. Then $G$ has at least one ...
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Why isn’t f(6Z + n) = 9Z + 2n a homomorphism from Z/6 to Z/9, even though g(n) = 2n is a homomorphism from Z to Z?

I was doing a problem, and I'm not sure as to why $f(6\mathbb{Z} + n) = 9\mathbb{Z} + 2n$ is not a homomorphism from $\mathbb{Z}/6\mathbb{Z}$ to $\mathbb{Z}/9\mathbb{Z}$, even though $g(n) = 2n$ is a ...
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Ideals of a field

I had the following - apparently straightforward - question on one of my past assignments: Show that a field has no other ideals except $\{0\}$ and the field itself. This was the proof I gave: ...
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Showing that stalk $O_{Spec \ A, p}$ is $A_p$

Suppose I have $A$, a commutative ring with unity. I would like to show that stalk $O_{Spec \ A, p}$ is $A_p$ for $p \in Spec \ A$. Could someone please explain me how this works? (I am having ...
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What do these terms mean: commutative, associative, distributive

I am reading a book, and I am trying to understand what the writer really mean by the following terms. I would like to understand what these words mean in relation to the examples. In regular ...
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Questions related to the concept of $k$-algebras

I am reading about modules and some days ago I've worked on some exercises related to $k$-algebras. The definition I've seen of $k$-algebra is that it is a field $k$ and a ring $A$ together with a ...
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Subgroups of $A_4$

So I have to show the 12 elements and arrange them into their cyclic subgroups. (I know there are $7!/2$, but I just have to show the cyclic subgroups.) That is my question. So I have: $e, $ ...
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Euclidean Domains in Ring Theory [closed]

Prove that $\displaystyle \mathbb{Z}\Bigg[\frac{1+\sqrt{-3}}{2}\Bigg]$ is a Euclidean Domain.
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Quadratic polynomial in ${\Bbb C}$ that vanishes on three different points of a complex line

Here is my question: Let $f(x,y)\in {\Bbb C}[x,y]$ be a quadratic polynomial. If $f$ vanishes on three different points (say, $p,q,r$) of a complex line $$ L:=\{(x,y)\in{\Bbb C}^2\mid ...
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Automorphisms in the Dihedral groups

Let g be a group and $a \in G$. Define $\phi_a:G\rightarrow G$ by $\phi_a(g)=aga^{-1}.$ Now Let $G=D_4$ and $a=r$, where $r$is the rotation. We must show that $\phi_r: D_4\rightarrow D_4$. So show ...
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Short exact sequence of modules

I am trying to show that if we have the following left splitting short exact sequence of $R-$modules: $0 \rightarrow M \stackrel{f} \rightarrow N \stackrel{g} \rightarrow S \rightarrow 0$ then there ...
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Give an example of a field where -1=1

The question is to find a counterexample to the following: In every field the element $-1$ is not equal to $1$. My intuition leads me to integers modulo $1$. Is this correct, are the integers ...
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Congruences in Algebra [on hold]

I have a question regarding a particular statement of a given Ring Theory problem. It is "$x$ is unique mod $n=n_1n_2...n_k$". Can anyone please tell me the meaning of this statement?
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Non-trivial centers, abelian towers, Lang

I am currently reading a proof in Lang's algebra that says that: because $G$ is a finite $p$-group, with non-trivial center "we have an abelian tower for $G/Z$ by induction, we can lift this abelian ...
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prove the following : [closed]

Prove that, if $K$ is a subgroup of $S_3$ generated by $(123)$, then every left coset of $K$ is also right coset of $K$.
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Polynomial decomposition

I've just recently learned about the neat algorithm that, given a polynomial $f$ finds (non linear) polynomials $h,g$ such that $$f = g \circ h \quad (1),$$ or decides that there are no such ...
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Calculate the degree of the field extensions.

I have been staring at this question for a while. I'm sure there is a little trick I am missing...anyway, it is the following: $ f = x^3 + x + 3 $ a) Show $f$ is irreducible over $\mathbb{Q}[x]$: ...
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Quaternion^2 in the symmetric group

Which is the minimum $n$ such as $Q_8\times Q_8\cong H$ with $H<S_n$? I now that the minimum n such as $Q_8$ embeds in $S_n$ is 8, so $Q_8\times Q_8$ embeds in $S_8\times S_8< S_{16} $, but is ...
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1answer
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Some questions about finite rings [closed]

Let $K$ be finite associative ring with nonzero multiplication. Are the following statements true: If for an element $a \in K$ there exist an element $b \in K$ such that $ax=b$ for all nonzero ...
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The krull dimension of $\Bbb{Z}$ and artinian rings

On page thirty of Matsumura, it says that $\Bbb{Z}$ has krull dimension 1 because every prime ideal is maximal. I understand this because for any prime p you have $0 \subset p$. However, for artinian ...
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The congruence $\{(a^m, a^{m+r})\}^\#$ on $a^+$.

I've spent a bit too long on this exercise. It's time to ask for help. This is Exercise 1.20 of Howie's Fundamentals of Semigroup Theory. Let $\rho_{m, r}$ (for $m, r\ge 1$) be the congruence ...
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Prove that the additive groups $\mathbb{Z}$ and $\mathbb{Q}$ are not isomorphic.

Is my proof below correct? What specific property of rationals did I exploit in my proof? It looks like the property I exploited is the following: Given any positive rational, I can always write it as ...
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Showing that a polynomial is irreducible

It seems intuitively clear that $xy-1$ in $\mathbb C[x,y]$ is irreducible. But I can't prove it rigorously. Could anyone show me how to prove it?
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Intermediate fields of Gal($x^5-2$, $\mathbb{Q}$)

I'm having trouble findind the fixed field of each subgroup of the Galois group of $\mathbb{Q(\alpha, \omega)}$, where $\alpha = 2^{\frac{1}{5}}$ and $\omega$ is a 5-th primitive root of unity. I list ...
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Novel approaches to elementary number theory and abstract algebra

As a part of a university course, I'll have to study Herstein's Topics in algebra and Hardy&Wright's Introduction to the theory of numbers. Can you suggest some books (to be used as companions) ...
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Abstract Algebra: Cyclic Groups (Lattice Diagram)

Example 4.2: Lets find all the subgroups of the given group and draw the lattice diagram for the subgroup. Z12 Z36 Z8 In the book finding the subgroups is explained well but it does not explain ...
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Why an $R-algebra$ requires $R$ to be commutative?

Here is the definition of $R$-algebra. Let $R$ be a commutative ring Let $(A,+\cdot)$ be an $R$-module. Let $\ast$ be a binary operation on $A$. If $x\ast(y+z)=x\ast y + x\ast z$ ...
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Lifting a direct summand of a free module

Suppose $R$ is a commutative ring, $I\subseteq R$ a principal ideal, and we're given split short exact sequences $ R \to R^n \to R^{n-1}$ and $ R/I \to (R/I)^n \to (R/I)^{n-1}$ the first inducing ...
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Herstein - Topics in Algebra - Polynomial rings page 157

In Chapter 3.9 of his book "Topics in Algebra" , 2nd ed, Herstein describes an example of a Quotient ring, namely $ F[x]/(x^3-2) = F[x]/A $ where $F = Q $ the rationals, and $(x^3-2) = A $ is the ...
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Natural isomorphisms of the forgetful functor

Let $U: \mathbf{Groups} \rightarrow \mathbf{Sets}$ be the forgetful functor. Must every natural transformation $\eta: U \rightarrow U$ be a natural isomorphism?