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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Show that any nontrivial ideal that is also a subspace of $M_{n\times n}$ over a field $F$ is maximal.

I just need to show that if $A$ is a nonzero square matrix, then I can "build" the identity matrix $I$, Then the ideal must be maximal.
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Extend isomorphism of subgroups to homomorphism of groups

Given two finite groups $G_1$ ang $G_2$ with respective subgroups $H_1$ and $H_2$ satisfying $H_1\cong H_2$ via the isomorphism $\phi$, is it always possible to extend $\phi$ to an homomorphism ...
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counter example for “every ideal is contained in a maximal ideal” in non-unital case?

As known, the fact "every ideal in a unital commutative ring is contained in a maximal ideal" is proven using Zorn's lemma, but it really uses that the ring has the identity. (While using Zorn's ...
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Solving System of Equations modulo a prime

Consider the equation: $$ C \equiv H.M.H^{-1} \pmod{p}, $$ where C, M H are, say, $2\times 2$ matrices, and $p$ is an odd prime. The elements of the matrices $C, M$ are all integers. The elements ...
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31 views

Existence of a non-abelian group of order $p^n$.

Question: Let $p$ be any prime and $n \geq 3$. Show that there exists a non-abelian group of order $p^n$. Attempt: Take $n = 3$. Writing $\mathbb Z_p \times \mathbb Z_p = \{e, \alpha_1, ...
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Categories for the working mathematician exercises III 1

I'm currently reading Mac Lane's Categories for the working mathematician and I'm having some trouble with the two following exercises from part III. Find (from any given object) an universal arrow ...
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1answer
60 views

Proving that a polynomial of the form $(x-a_1)\cdots(x-a_n) + 1$ is irreducible over $\mathbb{Q}$

I want to prove that for any set of distinct integers $a_1,\ldots,a_n$, the polynomial $$h = (x-a_1)\cdots(x-a_n) + 1$$ is irreducible over the field $\mathbb{Q}$, except for the following special ...
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For any subnormal subgroup of a finite group, must its normalizer be subnormal, too?

Let $G$ be a finite group and $H$ a subnormal subgroup of $G$. Must $N_G(H)$ be subnormal, too?
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1answer
23 views

Singular matrix with entries in a ring.

Given a matrix $M\in A^{n\times n}$, where $A$ is a commutative ring different from $\{0\}$, then we know that if there exists a vector $x\in A^n$ such that $Mx=0$, then $\det M$ must be a zero ...
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Isomorphism among quotient algebras

Let we have two different quotient algebras $ A/B$ and $M/N$ such that $B$ and $N$ are nilpotent ideals of $ A$ and $M$. I want to define an isomorphism $ \phi: A/B \to M/N$.Can you help me how I ...
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“Least upper bound property for a poset is equivalent to greatest lower bound property” vs existence of sup given inf

The least upper bound property says that, "Every nonempty subset of $A$ that $is$ bounded above has a least upper bound." The great lower bound property is defined similarly, and it's not difficult ...
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Associative Proof

I have a non-empty set $R$ with a binary operation $*$. If the pair has an identity element $e\in R$ and $$(a*b)*(c*d)=(a*c)*(b*d)$$ holds for all $a,b,c,d \in R$, how do I then prove that this is ...
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Why is $K^{\ast n}$ contained in the norm group?

http://www.bprim.org/cyclotomicfieldbook/rlmain.pdf In section 5, $K$ is a local $p$-adic field containing the $n$th roots of unity, and $L = K(\sqrt[n]{x} : x \in K^{\ast})$. Kummer theory tells us ...
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Some exercises from Introduction to Homological Algebra by J.J. Rotman (category) [on hold]

Please give solutions for these problems: Give an example of a covariant functor that does not preserve coproducts. Prove that every left exact covariant functor $T$: $_RMod$ → $Ab$ preserves ...
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2answers
26 views

Identity Element and Identity Properties [duplicate]

Learning more abstract algebra, really not the most enjoyable of subjects, as nothing seems all that clear cut, but here goes anyway. I have a set $\mathbb Q = \{{p \over q} : p,q\in \mathbb Z \text{ ...
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The intersection multiplicity of two coprime polynomials is less or equal than the multiplicity of their product?

Are given $H_1,H_2$ coprime polynomial of $K[X,Y]$ with $K$ a algebraic closed field, $P\in\mathbb{P}^2(K)$ a point. Holds that$$\mu_P(H_1,H_2)\leq m_P(H_1H_2)$$ where $\mu$ is the intersection ...
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Writing out a product of permutations as the product of disjoint cycles

let $ \tau = (1~3)(2~4) $ and $ \sigma = (1~2~4~5) $ Is it correct to say that (working from right to left) $ \sigma \tau = (1~3)(2~5) $ and $ \tau \sigma = (1~4)$ ?
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Is $\ker(\operatorname{nat}_H)=H$?

This question came in the exam today, sadly I couldn't answer it. The question said: Prove whether or not this is a true statement, stating the reason. $$\ker(\operatorname{nat}_H)=H$$ where ...
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64 views

Show $x^p-t$ has no root in the field $\mathbb{F}_p(t)$

I don't think I fully understand. Let's say there is a root $x_0 \in K=\mathbb{F}_p(t)$, where $p$ is a prime number. Then $x_0 = \frac{P(t)}{Q(t)}$ for some polynomials $P,Q \in \mathbb{F}_p[t]$. ...
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Fields in Abstract Algebra [duplicate]

How to prove the following: Show that $\mathbb Z_{n}$ is a field if and only if $n$ is prime.
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1answer
27 views

Proof of a result on closed subgroups of Galois group

Let $M\supseteq K$ be an algebraic normal extension. Then its Galois group is profinite. I have been told that in this hypothesis, if $H\leq G$, then $H''=H\iff \bar H=H$, where ...
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1answer
37 views

Conjugates of an $r$-cycle in $S_n$

How many conjugates does a cycle of length $r$ have in the permutation group $S_n$? I tried to find them but failed.
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Subgroup of group is normal [duplicate]

This question came in the exam today, unfortunately I couldn't answer it. The question said: Proof whether or not this is a true statement, stating the reason. Subgroup of group is normal I ...
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1answer
35 views

In $S_{3}$ what is the group generated by $(123)$?

In $S_{3}$ what is the group generated by $(123)$? Is there a way to find the elements of the group generated by $(123)$?
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1answer
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Show that an $R$-module homomorphism $\alpha:A \to B$ is injective.

I am working on an exercise on injective modules: Show that an $R$-module homomorphism $\alpha:A \to B$ is injective if the induced map Hom$_R(B,Q)\to$ Hom$_R(A,Q)$ is surjective for all injective ...
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Question about direct product of two groups.

Let $G=\mathbb{Z}_n \times \mathbb{Z}_m$ and $d=p^k$ for some prime $p$ such that $d$ divides both $n$ and $m$. Then $G$ has exactly $d\phi(d)+[d-\phi(d)]\phi(d)$. For example consider the group ...
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Understanding a quotient ring of continuous functions

In trying to understand another questions answer(to a question I asked), I realized that my fundamental lack of knowledge was in regards to the following question: In terms of functions, what does ...
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2answers
181 views

Divisor of a finite group

Suppose we have a finite group $G$ and $d\in \mathbb N$ is a divisor of $|G|$. We define the set $E_d= \{g\in G : g^d =1\}$. Prove that $d$ is also a divisor of $|E_d|$. So far I proved that ...
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How to learn commutative algebra?

I want to learn commutative algebra from scratch. I was wondering, as you guys are experts in mathematics, what you think is the best way to learn commutative algebra? Is there any video course ...
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Maximal ideals of the ring of all continuous functions [duplicate]

Let $R$ be the ring of all continuous functions from the interval $[0,1]$ to $\Bbb R$. For each $a\in[0,1]$ let $$A_a=\{f\in R \mid f(a)=0\}$$ Now firstly, this is part of an assignment problem, ...
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1answer
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The relationship of subnormal subgroups and modular subgroups of a finite group.

Let $G$ be a finite group, a subgroup $H$ of $G$ is called subnormal if it's a term of a composition series of $G$, and is called modular if it's a modular element of the subgroup lattice $L(G)$. My ...
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2answers
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Prove that a transformation of the identity functor of a Group $G$ (seen as a category) into itself is just an element of the center of $G$

I want to prove the follow: Suppose $G$ is a group seen as a category, prove that a transformation of the identity functor of $G$ into itself is just an element of the center of $G$. I'm not ...
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1answer
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Groups occuring as derived subgroups.

I want to prove this problem but I have no idea how to start it. If you know please hint me, thanks. Suppose that $G$ is a group that has subgroup which is cyclic, characteristic and not in the ...
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Factorisation of Characteristically Simple Group

Please am working on a project "Factorization of characteristically simple group". And it has been really difficult to locate articles relating to that, even on the web. So i want to know if any one ...
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1answer
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Find an ideal of $R = \mathbf{Z}/12\mathbf{Z} = \{0,1,2,\dots,11\}$ with two elements

Let $R = \mathbf{Z}/12\mathbf{Z} = \{0,1,2,\dots,11\}$. Find an ideal $I$ of $R$ which consists of two elements. How many elements does $R/I$ have? I thought the ideals would be $\{0\}$, ...
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1answer
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Show that the subset $\overline{I} = \{\overline{x}:x \in I\}$ is an ideal.

Assume that $I$ is an ideal of the ring $\mathcal{O}_d = \left\{ \begin{array}{ll} \mathbb{Z} [\sqrt{d}] & \text{ if } d \text{ is even } \\ \mathbb{Z} [ \frac{1 + ...
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3answers
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Show that if $A, B$ are groups, then $A \times B$ is solvable if and only if both $A$ and $B$ are solvable? [on hold]

Show that if $A, B$ are groups, then $A \times B$ is solvable if and only if both $A$ and $B$ are solvable? Why is obvious that if $A$ and $B$ are solvable then $A \times B$ is solvable?
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Proof that if $a,b \in G$ and $a^4b = ba$ and $a^3 = e$ then $ab = ba$

I tried to prove one of the examples in my Abstract Algebra book that stated: Prove that if $a,b \in G$ and $a^4b = ba$ and $a^3 = e$ then $ab = ba$ I went about just saying that $a^4b = ba ...
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2answers
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Prove that if $Z(G) = \lbrace x \in G: gx = xg \text{ for all }g\in G\rbrace$ then $Z(G)$ is a group

So my challenge is: Prove that if $Z(G) = \lbrace x \in G: gx = xg \text{ for all }g\in G\rbrace$ where $G$ is a group, then $Z(G)$ is a group Unlike this question: To show that the center is a ...
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Show that $f: \mathbb{Z} \to n\mathbb{Z}$ defined as $f(m) = mn$ is bijective

Let $n \in \mathbb{N}$. Show that $f: \mathbb{Z} \to n\mathbb{Z}$ defined as $f(m) = mn$ is bijective such that $f(m_1 + m_2) = f(m_1) + f(m_2)$, $\forall m_1, m_2 \in \mathbb{Z}$ To be bijective, ...
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Show that $x^{3}-3$ irreducible over $\mathbb{Q}(\sqrt{-3})$

Is there a slick way to show that $x^{3}-3$ is irreducible over $F= \mathbb{Q}(\sqrt{-3})$? What I did seems kind of convoluted (showing directly that there is no root in F). Thanks
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1answer
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algebraic integers of $\mathbb{Q}(\sqrt{d})$

Assume that $d$ is square-free. What is the set of algebraic integers in $\mathbb{Q} \left(\sqrt{d} \right) = \{a + b \sqrt{d}:a,b \in \mathbb{Q} \}$? The algebraic integers in ...
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$\gcd(a,n)\neq 1 \implies $ there is $b$ such that $ab\equiv 0 \pmod{n}$

I have that $\gcd(a,n)\neq 1$ ($a$ and $n$ are not coprime). Then, somehow, I need to prove that exists an $b$ such that $$ab\equiv 0 \pmod{n}$$ What I did: $$ab\equiv 0 \pmod{n}$$ is the same ...
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1answer
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Why does it mean that $n$-th variable is removable?

I'm reading the proof for "the fundamental theorem of symmetric polynomials" and I have a trouble with it (http://en.m.wikipedia.org/wiki/Elementary_symmetric_polynomial) Let $P(X_1,...,X_n)$ be a ...
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1answer
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Extension $\mathbb{Q}(\sqrt{-5}, i)/\mathbb{Q}(\sqrt{-5})$, splitting.

In the extension $\mathbb{Q}(\sqrt{-5}, i)/\mathbb{Q}(\sqrt{-5})$, must principal prime ideals of $\mathbb{Z}[\sqrt{-5}]$ necessarily split into 2? Must nonprincipal prime ideals not split? ...
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How to get real irreducible matrix representations from the complex irreducible matrix representations?

I'm trying to get real symmetry adapted orbitals for molecules with icosahedric symmetry (point groups $I$ and $I_h$) using the complete projector operator (truly projector if i=j): \begin{equation} ...
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2answers
46 views

Algebra and Maximal ideal.

I am trying to solve the following problem. If $ \mathcal{K}$ is a field and $a_1,a_2,\dots,a_n \in \mathcal{K}$. Prove that $(x_1-a_1,x_2-a_2,\dots,x_n-a_n)$ is a maximal ideal in ...
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1answer
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Non-abelian Group with infinite exponent in which every proper subgroup has finite exponent

can you find a Non-abelian Group with infinite exponent in which every proper subgroup has finite exponent?
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Fundamental theorem of Algebra using ideas of complex singularities

Below is an excerpt from Arnold's Theory of Catastrophes (I haven't got an American edition, so translating from Russian). Where I can read about it in more detail? Not only regarding polynomials. ...