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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Relation between Tensor-hom adjunction and adjugate matrix

Let $R\to S$ be a ring homomorphism, let $M,N$ be $S$-modules and $Q$ an $R$-module. Then, we have $$\textrm{Hom}_R(M\otimes_S N,Q) \cong \textrm{Hom}_S(M,\textrm{Hom}_R(N,Q).$$ I want to know ...
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Isomorphism as vector space or field

Q(√2) and Q(i) are isomorphic as vector spaces or as field. Which one is true ?
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Does the fundamental theorem of symmetric polynomials hold in any ring?

Fundamental theorem of Symmetric polynomials: Let $R$ be a commutative ring and $e_0,...,e_n$ be the elementary symmetric polynomials of $R[X_1,...,X_n]$. Let $\Phi:R[X_1,...,X_n]\rightarrow ...
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Noetherian ring under some conditions has atleast two minimal prime ideals

Question is : Suppose $R$ is a noetherian ring. Prove that $R$ is either an integral domain, has nonzero nilpotent elements, or has atleast two minimal prime ideals. [Use the previous exercise.] ...
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1answer
20 views

Irreducibility of a cubic polynomial

Let $f(x)=x^3+2x^2+x-1$. Then over which of the following fields $k$ is $f$ irreducible? $k=\mathbb{Q}$ $k=\mathbb{R}$ $k=\mathbb{F}_2$ $k=\mathbb{F}_3$ My Attempt: (2) $f$ is ...
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Is it customary to call it “R-module ring homomorphism”?

Let $R$ be a ring. Let $M,N$ be rings together with $R$-module structures, but $M,N$ are not $R$-algebras Let $\phi:M\rightarrow N$ be a ring homomorphism which is also an $R$-module homomorphism. ...
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1answer
24 views

Transitive action of the group implies isomorphism with the quotient by stabilizer

Let $\Omega$ be a set and $G$ a subgroup of the group $Sym(\Omega)$ of permutations of $\Omega$. Let $\omega \in \Omega$ and let $G_{\omega}$ denote the stabilizer of $\omega$ in $G$. If $G$ acts ...
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Show $f$ is irreducible.

Let $E$ be an extension field of $F$. Show that if $\alpha \in E$ is algebraic of degree $n$ over $F$ and $f\in F[X]$ is of degree $n$ with $f(\alpha) = 0$, then $f$ is irreducible. For this ...
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1answer
27 views

Every element of the form $x^n - \beta$ is a norm?

Let $F$ be a local $p$-adic field containing the $n$th roots of unity. The notes I'm reading claim that every element of the form $x^n - \beta$, for $x, \beta \in F$, is a norm from ...
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Show that there do not exist nonsingular matrices $P,Q ∈ M_{n×n}(F )$ satisfying $PAQ = A^T$ for all $A ∈M_{n×n}(F )$.

Let $F$ be a field and let $n$ be a positive integer. Show that there do not exist nonsingular matrices $P,Q ∈ M_{n×n}(F )$ satisfying $PAQ = A^T$ for all $A ∈M_{n×n}(F )$. If I set $A=I$, ...
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32 views

Number of elements of order $2$ in a group of even order [on hold]

If $G$ is a group of even order, then the number of elements of order $2$ in $G$ is: (a) 2 (b) 4 (c) even (d) odd
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1answer
23 views

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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3answers
24 views

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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1answer
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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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21 views

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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2answers
48 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
76 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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22 views

For any subnormal subgroup of a finite group, must its normalizer be subnormal, too? [on hold]

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
26 views

Singular matrix with entries in a ring. [duplicate]

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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18 views

Isomorphism among quotient algebras [on hold]

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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1answer
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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
27 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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35 views

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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1answer
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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
28 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
38 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
37 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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184 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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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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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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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 [on hold]

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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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
25 views

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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39 views

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 ...