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Questions tagged [abstract-algebra]

Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, and modules, among other topics. Whenever you use this tag, please also include related tags like group-theory, ring-theory, modules, etc., in order to clarify ...

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Are there any interesting algebraic structures where arbitrary infinite products are defined?

Are there any interesting instances of "super-semigroups", where we can use infinite products like: $b = (aaa\dots)$ $c = (ababab\dots)$ $d = (abcaabbccaaaabbbbccccaaaaaaaabbbbbbbbcccccccc\dots)$ ...
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When does Fermat's Last Theorem hold over finite fields?

It is well-known that in his attempts to prove Fermat's Last Theorem (FLT) over $\mathbb Z^+$, Schur came up with a result that has come to be known as Schur's Theorem, which implies that FLT fails ...
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The concept of associate for finite abelian groups

Let $G$ be a finite abelian group of order $n$, and let $a$ and $b$ be elements of $G$. If $a$ generates the same subgroup as $b$, must there be an integer $i$ prime to $n$ such that $ia=b$? I can ...
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Direct Product cancellation in Hopfian rings

A ring $R$ is called to be 'Hopfian' if every ring homomorphism of $R$ onto $R$ is an automorphism of $R$ Question: Given $R$, $S$, $T$, Hopfian rings and $$R \times S \cong T \times S$$ implies ...
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If $g:\mathbb{Z_{10}}\rightarrow U_{20}$ is a group homomorphism, then the order of $g(1)$ is either $1$ or $2$.

Why is that if $g:\mathbb{Z_{10}}$$\rightarrow$$U_{20}$ is a group homomorphism, then the order of $g(1)$ is either $1$ or $2$? Also, $g$ is a function, $\mathbb{Z_{10}}$ is the group of integers ...
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How do generators of a group work?

$G$ is a group, $H$ is a subgroup of $G$, and $[G:H]$ stands for the index of $H$ in $G$ in the following example: Let $G=S_3$, $H=\left<(1,2)\right>$. Then $[G:H]=3$. I know the definition of ...
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Showing that $(I+J)/(I\cap J)\cong (I+J)/I\times (I+J)/J$.

Show that $(I+J)/(I\cap J)\cong (I+J)/I\times (I+J)/J$, where $I,J$ are ideals in a commutative ring $R$ with identity. What I did was first show that $\phi: I+J\to (I+J)/I\times (I+J)/J$ defined by $...
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1answer
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The matrix corresponds to a homomorphism $\mathbb{Z^2} → \mathbb{Z^3}$

In Aluffi's Algebra in 4.3. Reading a presentation it says: For example, take $M=\begin{pmatrix}1&3\\2&3\\5&9\end{pmatrix}$; this matrix corresponds to a homomorphism $\mathbb{Z^2} → \...
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Showing that $M_n(R[x]) \cong (M_n(R)[x]$

I'm trying to show that $M_n(R[x]) \cong (M_n(R)[x]$ so I consider the mapping that sends an element $A \in M_n(R[x])$ to the polynomial whose coefficients are matrices in which the entries of those ...
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1answer
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Factorization Theorem

Let $f_1:A \longrightarrow A_1 , f_2 :A \longrightarrow A_2$ homomorphism of rings such that $ker(f_1)\subseteq ker(f_2)$. Prove that exist a homomorphism $f:A_1 \longrightarrow A_2 $ such that $f_2=f ...
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Representation Theory Direct Sum Composition

Let $A_5 \subset S_5$ be the alternating group on five letters. (a) Describe, as a direct sum of irreducible representations of $A_5$, the restrictions to $A_5$ of each of the irreducible ...
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Suppose that $H\trianglelefteq G$ and $H\leq K\leq G$. Show $H\trianglelefteq K$ and that $K/H\leq G/H$

This is not for an assignment. I found this practice problem in the back of my abstract algebra book and I'm trying to figure it in preparation for an upcoming exam. I've attempted the first part of ...
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Axioms of algebraic structure - ring

If in the definition of ring $(R,+,\times$) we insist that it has unit element $1$. Then we can show that addition $(+)$ is commutative operation. However, most of the proof which I've seen in MSE use ...
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1answer
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What is the cardinality of the following set?

What is the cardinality of the following set: $\{f: \mathbb{R} \rightarrow \{a+b\sqrt[3]4\ | a,b \in \mathbb{Q}\}\}$? Let $S = \{f: \mathbb{R} \rightarrow \{a+b\sqrt[3]4\ | a,b \in \mathbb{Q}\}\}$. ...
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Ordered subfields of $\mathbb{Q}_p$

I recently read about real ordered fields. Using real closures, I figured out that for each algebraically closed field $C$ of characteristic $0$, there exists a real closed subfield $R\subseteq C$ ...
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Conjugacy classes; Jordan Normal Form

Compute the conjugacy classes of the following groups: GL3(C);GL3(Z2);GL4 (R) I mean, do I have to use the Jordan Normal Form?
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1answer
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How to do congruence-class arithmetic?

When working through this question: Write out the addition and multiplication tables for the congruence-class ring $F[x]/(p(x))$. $F=\mathbb{Z_2}$; $p(x)=x^{3}+x+1$. [Question #1 in section 5.2: ...
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1answer
24 views

Structure of a group $G$ through its isomorphic images in $\operatorname{Sym}(G)$

Following the idea that a group is its structure, and reminding of Cayley theorem, I'm wondering whether we can virtually build up any finite group $G=\lbrace a_0,\dots,a_{n-1} \rbrace$ by searching ...
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1answer
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$M$ is an irreducible $R$ module $\iff$ $M$ is a cyclic module and every nonzero element is a generator

$M$ is an irreducible $R$ module $\iff$ $M$ is a cyclic module and every nonzero element is a generator. ($\rightarrow$) If $M$ is an irreducible $R$-module then it's obvious that $M$ is a cylclic ...
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2answers
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Proving the trace of a representation is equal to zero

I'm having some trouble in beginner's representation theory and am pretty lost about this problem: Let ($\rho$, $V$) be a representation of $G$, so $\rho$: $G$ $\to$ $GL(V)$ is a group homomorphism. ...
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Methods of finding the minimal polynomial of $w = e^{2i\pi/12}$. over $\mathbb{Q}$

So the minimal polynomial of $w = e^{2i\pi/12}$ will be given by the 12th cyclotomic polynomial, where the $n$-th cyclotomic polynomial can be calculated recursively by the following formula: $b=\...
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A Bound on the Dimensions of Certain Types of Subspaces

Let $V$ be a $4$-dimensional vector space over the complex numbers, and let $S$ be a subspace of the endomorphisms of $V$ such that the elements of $S$ commute. If there exists an element in $S$ ...
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Let $R$ be a comm ring w/ 1, $M$ be a left $R$-module. Show that $Hom_R(M,M)$, a left $R$-module

Let $R$ be a commutative ring with 1 and let $M$ be a left $R$-module. Show that $Hom_R(M,M)$, the set of $R$-homomorphisms from $M$ to itself is a left $R$-module if we set $(r·f)(m) = rf(m)$ for all ...
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3answers
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$\text{SL}_2(\mathbb Z)$ acts on upper plane $\mathbb H$. What kind of points have non-trivial stabilizer? And how many orbits are there?

$\text{SL}_2(\mathbb Z)$ acts on upper plane $\mathbb H= \{z \in \mathbb{C} | \Im(z) > 0 \}$ via Mobius transformation. $$ \text{ For } \gamma =\begin{bmatrix} a &b \\c&d \end{...
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Some counterexamples in basic ring theory

Give an example if possible, and if not possible explain why not. a) A subring of a PID that is not PID. b) A PID that is a subring of a non-PID. c) A subring of a PID that is not UFD. My approach:...
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2answers
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Prove that the following equation has no constructible solutions.

Prove that the following equation has no constructible solution: $\ x^3 - 6x + 2\sqrt{\pi} = 0$ The way I am trying to approach is that: I want to transform the equation into some integer ...
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About irreducible representations over the polynomial ring $k[x]$

From Example 2.3.14 (2) here page 21: Let $A = k[x]$. Since this algebra is commutative, the irreducible representations of $A$ are its 1-dimensional representations. They are defined by a single ...
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What is the relationship between the norm of a vector space and its linear functional operator norm?

I'm trying to answer the question below. I honestly cannot see any connection other than that both norm are measuring "size" in some way, which is just definitionally true. Let $V$ be a be a normed ...
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Explain why J is a prime Ideal in Z[x] [duplicate]

I am trying to prove that the ideal $J=<x+1>$ is prime in the ring $\mathbb{Z}[x]$. I know that if the generator is prime, then the ring modulo the generator is an integral domain. I can show ...
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27 views

Ring-like strusture with non associative addition

There is this structure i found which is the set of continuous maps from [-1, 1]^n into itself, endowed with a "sum" which is the pointwise sum of two functions divided by 2, and a "product" which is ...
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Why is the one-variable ring of polynomials with real coefficients a principal ideal domain? [duplicate]

I'm new to Ring Theory and I'd like to check if I'm on the correct track with this. Wolfram defines a PID as "an integral domain in which every proper ideal can be generated by a single element". As ...
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1answer
71 views

Is this a correct solution to show that the ring $\mathbb{R}[x]$ does not embed in $\mathbb{C}$

I know this is not a typical question and I will delete it later. Here is a question together my answer. Please let me know if my answer is correct. The reason that I am posting this is that I ...
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1answer
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Question about construction of The Grothendieck group.

In the Algebra by Serge Lang, he constructed a Grothendieck group of commutative monoid $M$, namely $K(M)$:(page 39-40) $M$ is a commutative monoid. Let $F_{ab} (M)$ be the free abelian group ...
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1answer
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When is an element of an extension field in the base field?

Given a finite field $\mathbb{F}_p$, consider an extension of it; $\mathbb{F}_{p^m}$. If I'm given $\alpha \in \mathbb{F}_{p^m}$, then, if $\alpha^p = \alpha$, $\alpha \in \mathbb{F}_{p}$. Why is ...
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2answers
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Proving polynomial over Q is irreducible

Prove that if $a$ and $b$ are odd then the polynomial $$x^3+ax+b$$is irreducible over $\mathbb{Q}$ I would be very much thankful if someone could help me with this one.
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1answer
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What does $φ(a) = a$ mean in this statement?

Let $F$ be a field and let $φ:F[x] \to F[x]$ be an isomorphism such that $φ(a)=a $ for every $a$ in $F$. Prove that $f(x)$ is irreducible in $F[x]$ if and only if $φ(f(x))$ is. [Hint: First prove that ...
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1answer
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Integral Closure, Galois extension,and Dedekind Domain

Let $A$: Dedekind domain, $K$: $\operatorname{Frac}(A)$, $B$: Dedekind domain with $A \subset B$, $L$: $\operatorname{Frac}(B)$ Let $L/K$: galois extension with galois group: $G$. $B^G=\{b \in B \...
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Prove U=Z where addition is regular addition in Z, but multiplication is defined by a*b=a IS NOT A RING

Ok,so clearly a+b ∈ R a+(b+c)=(a+b)+c a+b = b+a a+0 = a = 0+a, 0 ∈ R a ∈ R a+x = 0, x = -a ∈R now a ∈ R b ∈ R and ab = a ∈R a(bc) = a(b) = a (ab)c = (a)c = a a(b+c) = ab+ac = a+a (a+b)c = ac+bc =...
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1answer
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Every Sylow subgroup is normal, then $G$ has a subgroup of order $m$ for every division $m$ of $|G|$

Some trouble working out an algebra problem. Suppose that every Sylow subgroup of a finite group $G$ is normal. Prove that $G$ has a subgroup of order $m$ for every divisor $m$ of $|G|$.
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1answer
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Check whether a subset of a vector space is a subspace

$V = \mathbb{K}^n$, where $\mathbb{K}$ -- field. $V_1 = \{(x_1,\cdots,x_n)\in V\mid \sum_{i=1}^{n} a_ix_i = 1; a_1,\cdots a_n \in \mathbb{K}\}$. So, I should check that $V_1$ -- subspace. At this ...
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1answer
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If $G$ is a finite simple group and $n=[G:K]>1$, then $|G|$ is a divisor of $n!$. [on hold]

If $G$ is a finite simple group and $n=[G:K]>1$, then $|G|$ is a divisor of $n!$. This is the fourth question in a series of exercises and I cannot figure out a solution nor why it belongs with ...
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Background of Commutative Algebra for Cohen-Macaulay orders and bibliography

In this semester, I'm doing a project in Algebra, and I would like take some advices and suggestions. To be more precise, I will study the Cohen-Macaulay Orders and modules in relation to the Krull ...
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1answer
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Determining ideals, isomorphic rings of $\Bbb C[x, y]/(y^2 - x^3)$?

I've been having a substantial amount of trouble trying to understand the workings of $\Bbb C[x, y]$ mod... anything really. I figure this particular example is a good one to ask here because I ...
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1answer
42 views

Showing an isomorphism between two quadratic rings

Let $a,b$ be squarefree integers and set $R = \mathbb{Z}[\sqrt{a}]$ and $S = \mathbb{Z}[\sqrt{b}]$. Prove that 1) There is an isomorphism of abelian groups $(R,+) \cong (S,+)$. Let $\varphi : R \to ...
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2answers
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Ideals generated by two elements in $\mathbb{Z}[x]$

Consider the following ideal $(2+x,x^2+5)$ in $\mathbb{Z}[x]$. Then I showed that $(2+x,x^2+5)=(9,2+x)$. But I am not able to do the same for ideals $(1-4x,x^2+5)$ and $(1+2x,x^2+5)$. Is there some ...
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1answer
34 views

Prove operator norm cannot be longer than len(basis) times max(norm(basis))operator norm

Let $V$ be a finite dimensional normed linear space and let $T \in \mathscr L (V)$. Define the operator norm of $T$ to be the smallest number $M$ such that $||T v|| ≤ M||v||$ for any $v \in V$ . We ...
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The cases in proving that a group of order 90 is not simple

I am trying to attempt this problem, but I am wondering why exactly these are the two cases the problem is split into. I can understand the first case, since that lets us count elements and get a ...
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22 views

show that the order of C(a) must be odd

I am working on the following problem from group theory: If $n$ is odd and $a\in S_n$ is an n-cycle, $a=(a_1,a_2,......,a_n)$, show that no element of the centralizer $C(a)=\{g\in S_n \mid ga=ag\}$ ...
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2answers
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Showing that there is a surjective map from $\Bbb Z \ast \Bbb Z$ to $C_2 \ast C_3$ just using universal property of coproduct

I am solving Allufi chapter $0$ exercise $3.7$. There is a easy way to solve this if we know how the coproduct of $\Bbb Z \ast \Bbb Z$ and $C_2 \ast C_3$. I was wondering if there is an abstract ...
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22 views

Existence of ring homomorphism and embedding

I don't know how to prove the following: Let $K\subset E$ be a field extension. i) Let $x$ be an element of $E$. Show that $x$ is transcendent over $K$ if and only if the map $ev_{x}: K[X] \to E$ ...