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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Solve the equation f(f(x))=x^2+x

Solve the equation $ f(f(x))=x^2+x$, where $f$ is any function. If $f(x)=x^u+x$, $f(f(x))=(x^u+x)^u+x$ $\rightarrow$ $(x^u+x)^u=x^2$. I don't know what to do next...
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Gaussian integers and polynomial ring

I was reading some posts regarding the quotient Gaussian integers, and am now confused about how a quotient Gaussian integers can be represented/explained in the way of a polynomial ring (or a ...
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What is the pure essence of a definition of semantics?

What is the very essence of the definition of semantics (and interpretation, structure, model) for a logician or for an algebraist? We all know the usual definitions. But is it not the essence of ...
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30 views

Proving certain groups are normal

Given the following subgroup of $\mathbb S_4$: $$K=\{\mathrm{id},(12)(34),(13)(24),(14)(23)\}$$ prove that $K \unlhd \mathbb A_4$, $K \unlhd \mathbb S_4$ I am trying to solve this problem doing as ...
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44 views

Are there counter intuitive interpretations of ZF or ZFC?

Are there interpretations of ZF or ZFC that are non standard in the sense that $\epsilon$ is interpreted in a counter intuitive way that intuitively has nothing to do with "belongs to" or "is part of" ...
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73 views

Paradox in ring theory — what am I missing?

I saw somewhere the following exercise: Give example of prime ideal in a ring which is not maximal the answer was this: Let $R$ be our Ring and $I$ ideal such $$ R = {Z}[{X}] $$ $$ I = (x) $$ ...
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2answers
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An elliptic element of $SL(2,\mathbb{R})$ conjugate to a rotation

Show that an elliptic element of $SL(2,\mathbb{R})$ is conjugate to a rotation. An element $A$ of $SL(2,\mathbb{R})$ is called an elliptic element if $|\text{tr}(A)|<2$ As $|\text{tr}(A)|<2$ ...
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14 views

Finding commutator of $\mathbb D_n$ and $\mathbb S_4$

I have to find $[G,G]$ (the commutator of $G$) for the the dihedral group, $\mathbb D_n$ ; and the symmetric group, $\mathbb S_4$. What tools, theorems could I use in order to find these two ...
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59 views

There are no field structures on $\mathbb{R}^3$, but what of $\mathbb{R}^n$ for $n\geq 4$?

Has it been proved that there do not exist nice field structures on $\mathbb{R}^n$ for $n\geq 4$? The quaternions $\mathbb{H}$ fail due to lack of commutativity and the bicomplex numbers ...
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54 views

Is there a theory of “rings” with partially defined multiplication?

Consider the abelian group $R [[\mathbb{Z}^d]]$ of all formal Laurent series over a commutative ring $R$ (a typical element has the form $\sum_{v \in \mathbb{Z}^d} \lambda_v \cdot X_1^{v_1} \dots ...
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left regular representation of a group thru group action

Let G be a group and let $g\cdot:G\to G$(i.e.$g\cdot g'=gg'$) . This induces a permutation representation of the group. I was trying to walk thru the problems in dummite and foote. One of the problem ...
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Number of non isomorphic groups of order $122$, My attempt through Sylow theory.

$|G|=122 = 2 . 61$ No. of sylow $2$ subgroups $= 1$ or $61 = n_2$ No. of sylow $61$ subgroups $= 1 = n_{61}$ Let the group of order $61$ be $H_{61}$ and the group of order $2$ be $H_2$ Then : ...
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Proper Field extensions

Given a field $F$, is there a proper field extension $K$ such that any root in $K$ of a polynomial in $F[X]$ is in $F$? Note: I am not looking for the algebraic closure of $F$. One candidate is the ...
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26 views

Rationale for expressing as a direct sum and a direct product

In "Ireland and Rosen" page 35, it says if $R_1, R_2, ..., R_n$ are rings, then $R_1 \oplus R_2 \oplus \dots \oplus R_n = S$ is the direct sum of the $R_i$. Later in a proposition it says if $S = ...
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Is there some efficient way besides Eisentein's criterion to show that polynomials are irreducible?

While solving some problems, I had to show that $y-x^2$, $y^2-x^3$ are irreducible in $k[x,y]$ ($k$ is an algebraically closed field). The Eisenstein criterion don't apply here. Is there some other ...
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1answer
86 views

Prove that the ring of rational numbers $\Bbb Q$ is not isomorphic to the ring of real numbers $\Bbb R$

just wondering if my reasoning is correct. I said assume there is such a homomorphism f, then f(1)=1 since it is a ring homomorphism. But $$f(\sqrt 2)= f(1\cdot\sqrt 2)= f(1) \cdot \sqrt 2= \sqrt ...
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1answer
55 views

What can you say about the order of this element?

I am self-studying algebra and encountered the following problem: If $b^{-1}ab=a^2$ and $c^{-1}ac=a^3$ and $b,c$ has orders $4$ and $3$ respectively, what can you say about the order of $a$? In ...
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1answer
31 views

On finding the number of homomorphisms from $G$ to $G_1\oplus \cdots \oplus G_n$

How shall I establish that the number of homomorphisms from the group $G$ to $G_1\oplus G_2\oplus \cdots G_n$ is same as $h_1h_2\cdots h_n$ where $h_i$ is the number of homomorphisms from $G$ to $G_i$ ...
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1answer
18 views

Prove that the Galois group of a polynomial $p(x)=x^q-1$ is the Cyclic group of order $q-1$, where $q$ is prime.

I understand why $q$ must be odd, since complex conjugation must be one of the Galios group's elements. But why must $q$ be prime?
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Example of ring which is neither commutative nor unital

Give an example of ring which is neither commutative nor unital. I think, subring of matrix ring is neither commutative nor has a unit element.
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32 views

Clarify Cartesian Products and Binary Operations

So tell me if I'm saying this write. A Cartesian Product is a function f:X x Y --> Z , where some unknown structural operation on the sets X and Y produces a set Z as its codomain, and Z is a set of ...
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19 views

Separable polynomial and algebraic extension

If $f\in F[t]$ is separable and $E/F$ is an algebraic extension, then how can I be sure that $f$ is separable as an element of $E[t]$? I thought it is a trivial question...but now I think it is ...
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Why should I consider the components $j^2$ and $k^2$ to be $=-1$ in the search for quaternions?

I'm reading a paper about Hamilton's discovery of quaternions and it explains why he failed in his 'theory of triplets' where he tried to make a vector with $3$ dimensions, as an analogy to the ...
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29 views

How to prove cyclic here? [duplicate]

Let $G$ be a finite group with identity $e$. Suppose that for every positive integer $d$, the number of elements $x\in G$ with $x^d = e$ is at most $d$. Show that $G$ must be cyclic.
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1answer
38 views

Explain rings and is [S, /, -] a ring?

Okay, so we are going to use the base set of numbers [i], which contains all possible cases of ai, where a is any real number. Here are 4 possible groups on this set --> [i,*]... [i,+]... [i,/] ...
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47 views

Does every free $R$-module have a maximal proper submodule?

Let $R$ be a commutative ring with $1$. We know that every finitely generated $R$-module has a maximal proper submodule. Is it true for any free $R$-module? In particular, can we do the following: ...
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Size of a subset of the set of units of a quotient ring

Let $R$ be a commutative Dedekind domain with multiplicative identity $1$, let $k$ be a positive integer, and let $I$ be a nonzero proper prime ideal of $R$. Is there a way to find the size of the set ...
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105 views

Cardinality of $\text{Aut}(G\times G) $

Let $G$ be a finite group. If $|\text{Aut}(G)|$ is known, what can we say about $|\text{Aut}(G\times G)|$ ?
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1answer
25 views

Solvable implies quotient group is solvable: Proof check.

I'd like to check the veracity of my proof. I've seen several proofs using different methods (some I'm allowed to use with lots of element-pushing and others using ideas I'm not allowed), but none ...
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1answer
74 views

How to find $10^{-1}$ in $\{0, 2, 4, 6, 8, 10, 12\} \subseteq \mathbb Z_{14}$?

Here is my question: Consider the subset $S = \{0, 2, 4, 6, 8, 10, 12\}$ in $\mathbb Z_{14}$, with the operations of addition and multiplication in $\mathbb Z_{14}$. (a) Show that $S$ has ...
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Product in the category of pointed sets..

I have the category $C$, where: objects are nonempty sets with one fixed element $Obj = \{(A,a)$, where $A$-nonemty sets, $a\in A\}$, morphisms are $Mor=\{ f:(A,a)\rightarrow (B,b)$; where $f$ - is ...
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Find a generator for an ideal in $\mathbb{Q}[T]$

Let $I$ be the ideal in $\mathbb{Q}[T]$ generated by $L=\{T^{2}-1, T^3-T^2+T-1,T^4-T^3+T-1\}$. Find $f\in\mathbb{Q}[T]$ such as $(f)=f\mathbb{Q}[T]=I$. The book solution proves that $I\subseteq ...
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Fields - Proof that every multiple of zero equals zero

This is one of my first proofs about fields. Please feed back and criticise in every way (including style and details). Let $(F, +, \cdot)$ be a field. Non-trivially, $\textit{associativity}$ implies ...
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1answer
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If $\mathbb{C}[x,y]/I$ is finite dim $\mathbb{C}$-vsp, does it have a monomial basis? Related to Hilbert Scheme of points in the plane.

Background (You can skip ahead if you wish): I'm trying to read this article about the Hilbert Scheme of points in the plane, and I don't understand one of the claims. An ideal $I\subset ...
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1answer
27 views

Extension of an $R$-homomorphism as a sum

I want to solve this problem: Let $M$ be an injective module over a ring $R$ with identity, and $f$, $g$, $h$ are elements of the ring of $R$-endomorphisms $\operatorname{End}(M)$ such that $\ker ...
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48 views

Conditions for a group to be isomorphic to semidirect product of its subgroups

Let $G$ a group and $N$ a normal subgroup of $G$. If $G$ it have a subgroup $H$ s.t. $H \cap N $ is the trivial subgroup and $H$ is isomorphic to $G/ N$ then $G$ is isomorphic to $N\rtimes H$. Could ...
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1answer
47 views

Proof of sylow first theorem

Im studying the proof of sylows first theorem (from abstract algebra by beachy and balair). I have 2 questions about it, I will present the proof below: (theorem: Let $G$ be a finite group, if $p$ is ...
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Morandi's Rings appendix: about a step of the proof that that $R[x]$ is a UFD if $R$ is. [duplicate]

In the appendix about rings in Patrick Morandi's book Fields and Galois Theory, we find the following exercise (which arises in the proof of the theorem: $R[x]$ is a UFD if $R$ is a UFD). Let B be ...
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1answer
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Alternating group $A_n$ generated by its subgroups $A^{(i)}_{n-1}$

Suppose $A^{(i)}_{n-1}$ is the alternating group taking the $n-1$ numbers $\{ 1, 2, \cdots, i-1, i+1, \cdots, n \}$ that are the domain of even permutations in it, where $n \geq 4$ and $i =1, 2, ...
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Question about the Averson's proof of the bicommutant theorem.

In the Averson's proof of the bicommutant theorem is proved that, if $A$ is a self-adjoint algebra of operators with trivial null space and $T \in A''$, for every $\epsilon>0$, $n=1,2..$ and every ...
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1answer
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Something that is true for every element of $\text{Sym}(\Bbb{N})$

I'm trying to prove that: Every element of $\text{Sym}(\Bbb{N})$ can be written as $f\circ g$ for some $f,g\in \text{Sym}(\Bbb{N})$ with $f^2=g^2$. But I can't even prove this for ...
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1answer
130 views

Proof that group is commutative if every element is its inverse (feedback wanted)

This is one of my first proofs about groups. Please feed back and criticise in every way (including style & language). Axiom names (see Wikipedia) are italicised. $e$ denotes the identity element. ...
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Subring of $\mathbb{Z}[i]$ and an infinite set $X$ such that $\exists x \forall y \in X \,\,x^2\mid y^2$ but $\forall x \forall y \,\,x \not\mid y$

This is a question derived from A subring of the ring of Gaussian integers such that $a^2 \mid b^2$ does not lead to $a\mid b$ in infinitely many such cases. Is there a subring $R$ of Gaussian ...
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44 views

A question on short exact sequences.

The following is an excerpt from Atiyah-Macdonald on short exact sequences. I don't understand the part where the author says "Then $d(x'')$ is defined to be the image of $y'$ in Coker ($f'$)". Is ...
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Compound functions: one to one and onto

Let $f: A \to B$ and $g: B\to C$ be maps. If $g(f(x))$ is one-to-one and $f$ is onto, show that $g$ is one-to-one I'm really not sure how to prove this. Would someone be able to walk me through ...
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Determine if these correspondences on ${\mathbb Q}$ define functions

Given the correspondence $f: \Bbb Q \to \Bbb Q$, explain why $f$ is a function: a) $$f\left( \frac pq\right) = \frac {3p}{3q} $$ b) $$f\left( \frac pq\right) = \frac {3p^2}{7q^2}- \frac pq $$ ...
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47 views

A subring of the ring of Gaussian integers such that $a^2 \mid b^2$ does not lead to $a\mid b$ in infinitely many such cases

As the ring of Gaussian integers is a UFD, this means that $a^2 \mid b^2$ leads to $a\mid b$. Is there any subring of the ring of Gaussian integers with infinitely many elements such that ...
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26 views

regular representation of algebras

Let suppose we have universal enveloping algebra, what is the meaning of the notion of the right regular representation of that? How can we determine the right regular representation of universal ...
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61 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 $f(x) = x^n-x = x(x^{n-1}-1)$ and $f'(x) = nx^{n-1}-1$ If $f(x)$ has a multiple zero, then, $f(x)$ and $f'(x)$ have a common factor. ...
3
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28 views

Question about a passage in the Bicommutant Theorem's proof.

In the Averson's book, in the proof of the Von Neumann's Bicommutant theorem there is this passage: ($A $ is a self-adjoint algebra of operators in $L(H)$) "Let $\xi_1$ be an element of the Hilbert ...