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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Ideals of the residual classes $\mathbb Z_n$

Let $n$ be a positive integer and considere the ring $\mathbb Z_n$ of the residual classes modulo $n$. My intuition tells me that there is no two distinct ideals of $\mathbb Z_n$ with the same number ...
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Characterising subgroup

Let $\omega $ be a path in $\hat{X}$ with $\omega(0), \omega(1) \in p^{-1}(x_0)$, where $p$ is a covering map $p:\hat{X} \rightarrow X$. Let $\alpha=[p \circ \omega] \in \pi_1(X,x_0)$. Then we have ...
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Finding this Polynomial Subspace

Let $A = k[x^{\pm 1}, y^{\pm 1} ] $, considered as a $k$ - algebra. Can someone give me a nice description of the (vector) subspace: $$ A_0 = \lbrace (f,g) \in A^2 : \frac{ \partial f}{\partial y} = ...
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1answer
42 views

Cohomology group $H^n(G, \mathbb{R}/\mathbb{Z})$ as a continuous group or a Lie group?

Is there any example of Borel-Moore cohomology group $H^n(G, \mathbb{R}/\mathbb{Z})$ for any $G$ such that $H^n(G, \mathbb{R}/\mathbb{Z})$ is a continuous group? Such as a Lie group? Most of the ...
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co and contravariant vectors, their difference and properties

Very often when talking about covectors, co- and contravariant stuff, it's mentioned that there is no difference in "normal" linear algebra. That the difference only comes "when dealing with curved ...
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1answer
39 views

Proving that a field $K$ can be generated by algebraically independent elements and an separable element

Let $k$ be a perfect field (either $k$ has characteristic $0$, or characteristic $p > 0$ and every element has a $p$th root), and let $K$ be a finitely generated extension field. I have a question ...
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1answer
22 views

Is a torsion-free discrete abelian group of finite rank isomorphic to a subgroup of $\mathbb{Q}^k_d$?

If $G$ is a torsion-free discrete abelian group of rank $k$, then is it true that $G$ is isomorphic to a group $H$, where $\mathbb{Z}^k < H < \mathbb{Q}^k_d$? Here, $\mathbb{Q}^k_d$ is the ...
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Irreducibility in Galois/non Galois Extensions

Let $k$ be a field and $\alpha$ algebraic over $k$. Let $K$ be the Galois closure of $k(\alpha)$ (obtained by adding all conjugates of $\alpha$). If $f(x) \in k[x]$ is irreducible over $k[\alpha]$ is ...
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Projective special linear groups

Is it known if $PSL(2,\, F)$, with $F$ a field of prime $p$ characteristic (maybe with all proper subfields of finite order), is co-hopfian? I've searched everywhere but found nothing. Definition A ...
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In an Integral Domain, every prime is an irreducible. Flaw in the Proof?

In an Integral Domain, every prime is an irreducible. The proof is as follows : Let $D$ be the integral domain, then, if $a \in D$, it's possible to express a = $bc$ where $b,c \in D ...(1)$. ...
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29 views

When a system of rational linear homogeneous equations have complex solutions

Question: When a finite system of rational linear homogeneous equations in finitely many variables have a nontrivial complex solution (that is not a rational solution), does it imply that there is ...
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43 views

Additive inverse in a field is unique

there will be an $a$ let's prove $-a$ is unique. let's assume there are 2 additive inverses $b$ and $c$ therefore $a+b=a+c$ let's multiply them by the multiplicative inverse of $a$ I try to use the ...
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1answer
14 views

Let x, y be integers. Show that if x = y (mod n), then x + mZ = y+mZ,and conversely, if x+mZ=y+mZ then x = y (mod n)?

Let $x, y$ be integers. Show that if $x = y\mod n$, then $x + nZ = y+nZ,$ and conversely, if $x+nZ=y+nZ$ then $x = y\mod n$? I have no a clue on how to prove this! Please help.
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1answer
36 views

Show that the set $U_{77,76}$ of solutions to $x^{76} = 1$ is a subgroup of $U_{77}$.

Show that the set $U_{77,76}$ of solutions to $x^{76} = 1$ is a subgroup of $U_{77}$. I don't understand this question at all but could someone also explain what it means by $U_{77}(76)$.
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1answer
37 views

A Fixed Field of $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$

I was working on a review problem from Dummit and Foote and came across the following issue. It is clear that the Galois group of the splitting field for the polynomial $(x^2-2)(x^2-3)(x^2-5)$ has ...
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If $G$ is a non-cyclic group of order $n^2$, then $G$ is isomorphic to $\mathbb{Z_n} \oplus \mathbb{Z_n}$

I've independently come up with a question (I know it's been asked before, but I can't find the question online) involving the external direct product, non-cyclic groups and isomorphisms. So, is the ...
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106 views

What is subtraction?

Let $a, b \in \mathbf{R}$. It is an elementary fact that addition is a commutative binary operation on the reals, that is, $a + b \in \mathbf{R}$ and $a + b = b + a$. With the exception of ...
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41 views

Recovering Unique Factorization

Can a (commutative) ring $S$ fail to have unique factorization but be a subring of a (possibly noncommutative) ring $R$ which does have unique factorization? The idea being that the irreducibles in ...
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Determinant of the linear map given by conjugation.

Let $S$ denote the space of skew-symmetric $n\times n$ real matrices, where every element $A\in S$ satisfies $A^T+A = 0$. Let $M$ denote an orthogonal $n\times n$ matrix, and $L_M$ denotes the ...
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Ideal in a certain algebra over a field

Let $K|k$ be a finite field extension. Define $D$ to be a finite dimensional $k$ division algebra. If $J$ is a nonzero two-sided ideal of $D\otimes_k K$ then by considering $K$-dimensions, I see that ...
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Presentation of an object in an Eilenberg Moore category by generators and relations

Let $\cal C$ be a category and let $T \colon \cal C \longrightarrow C$ be a monad. An $T$-algebra $A$ is presented by generators $G \in \cal C$ and relations $R \in \cal C$ if it is the coequaliser of ...
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1answer
42 views

Relation of order of a permutation with its sign

Let $G$ be a group with order $2m$ where $m$ is odd. Consider the left action $\lambda_g:G\to G$. It appears that if $g$ has odd order iff $\lambda_g$ has odd order iff $\lambda_g$ is an even ...
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Is $\mathbb{C}[x,y] / (y^2-x^3)$ a PID?

First, I'd like to show $\mathbb{C}[x,y] / (y^2-x^3)$ is an integral domain. Then I need to find out whether or not it is a PID. For the first part, I want to show $y^2-x^3 \: | \: fg \implies ...
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54 views

module over a quotient of a principal ideal domain

The Statement I suspect the following proposition is well known, but I found no reference. Proposition If $A$ is a principal ideal domain, if $I$ is a nonzero ideal of $A$, and if $M$ is an ...
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1answer
46 views

Powers of Orbifold Fundamental Groups

I have reduced a problem to $\pi(Y)^n/G^n$ where Y is a manifold and G is a group acting on the manifold. Can I "factor out," the $n$? i.e. $(\pi(Y)/G)^n$. Note that $\pi(X)$ is the fundamental group ...
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1answer
14 views

Prove $x_1$ is at least a $k$-fold root of polynomial $p$ if and only if $p(x_1) = p^{'}(x_1) = \dots p^{(k-1)}(x_1) = 0$?

Suppose $p: \mathbb R \rightarrow \mathbb R$ is a polynomial given by $x \mapsto a_nx^n + \dots a_1 x_1 + a_0$. How do I prove $x_1$ is at least a $k$-fold root of $p$ if and only if $p(x_1) = ...
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1answer
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Tangent line at $x_1$ to polynomial curve $p(x)$ of degree at least $2$ implies $x_1$ is a double root of $p(x) - p^{'}(x_1)(x-x_1)$?

Tangent line at $x_1$ to polynomial curve $p(x)$ of degree at least $2$ implies $x_1$ is a double root of $p(x) - p^{'}(x_1)(x-x_1)$ ?. Suppose I have a polynomial function $p(x): \mathbb R ...
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Find the $4$ sq. roots of $100$ in $ U_{209}$. Identify which square root of $100$ is square.

Find the $4$ sq. roots of $100$ in $U_{209}$. Identify which square root of $100$ is square. (Not the $4^{th}$ root, but the $4$ square roots). I honestly don't even know what this question is ...
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1answer
33 views

Product of divisible module is divisible

I have the following problem Is the product of divisible $R$-modules divisible? I think it is not. But I need some counterexample to this, somebody can give me a "place" to search one? Thanks a ...
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2answers
53 views

Dimension of $\dim_{\mathbb C}\mathbb C[X,Y]/I(Y^2-X^2,Y^2+X^2)$

I'm trying to solve the question 1.36 from Fulton's algebraic curves book: Let $I=(Y^2-X^2,Y^2+X^2)\subset\mathbb C[X,Y]$. Find $V(I)$ and $\dim_{\mathbb C}\mathbb C[X,Y]/I$. Obviously ...
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35 views

“The regular languages over $A$ are the homomorphic pre-images in $A^*$ of subsets of finite monoids.”

I'm trying to understand the statement: The regular languages over $A$ are the homomorphic pre-images in $A^∗$ of subsets of finite monoids. which appears in the Wikipedia article on free ...
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1answer
47 views

Tensor products and isomorphic algebras

I found that $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \simeq \mathbb{C} \oplus\mathbb{C}$ and that $ \mathbb{H} \otimes_{ \mathbb{R}} \mathbb{C} \simeq M_2( \mathbb{C})$. Could anybody hint me how ...
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A finite commutative ring with 1 whose elements satisfy a particular equation

I would be very grateful if you give me a hint on it: Suppose $R$ is a finite commutative ring with identity such that $ x^3 = x $ for all elements $x$ of $R$. Then $R$ is a finite direct product ...
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1answer
38 views

Irreducible module

I've met the following definition of irreducible module: an $R$ module $M$ is said to be irreducible if it contains no proper submodules: in other words, if $N \subset M$ is a submodule than either ...
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1answer
60 views

Universal property of polynomial ring in $\mathbf{CRING}$

I know that the polynomial ring $A[x]$ is the free $A$-algebra on $\{x\}$; this is its universal property in the category of $A$-algebras. Is there also a universal property for $A[x]$ considered as a ...
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1answer
31 views

When a subgroup of automorphism group of a structure is in the form of automorphism group of a substructure?

Question 1: Is the following statement true? ($*$) Let $\mathcal{L}$ be a first order language and $\mathcal{M}$ a $\mathcal{L}$-structure and $H\leq Aut(\mathcal{M})$ then there exists a ...
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2answers
128 views

Isomorphism of $(\Bbb{Z}, *)$ and $(\Bbb{Q}-\{0\}, \cdot)$

Are there any operations on $\Bbb{Z}$ that makes it isomorphic to $(\Bbb{Q}-\{0\}, \cdot)$ as a group? Edit: the operation should be made of addition and multiplication of integers, possibly ...
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1answer
36 views

Third point of elliptic curve $E: y^2 = x^3 + Ax + B$ given points $P_1=(x_1,y_1), P_2=(x_2, y_2)$ on $E$ (Weierstrass equation).

Third point of elliptic curve $E: y^2 = x^3 + Ax + B$ given points $P_1=(x_1,y_1), P_2=(x_2, y_2)$ on $E$ (Weierstrass equation). Assume $x_1 \neq x_2$. I create the straight line $y = m(x-x_1) ...
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Prove a property about the centralisator

Let G be a group and $U \subseteq G$ a subgroup. Let $x \in G$ be arbitrary. How to show that $C_G(xUx^{-1})=xC_G(U)x^{-1}$ where $C_G(U):=\{g\in G : gu=ug$ $\forall u\in U\}$ For the first ...
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Representing multivariable polynomials as matrices

Is there a nice way to represent polynomials in $x$ and $y$ of degree say $n$ as matrices, so that multiplication works out in a nice way? Maybe a ring homomorphism or something? I'm sorry that this ...
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77 views

The ring $R$ is a field iff $R$ has a finitely generated divisible module

I have the following problem: Prove that a integral domain $R$ is a field iff there exist a finitely generated divisible $R$-module. One side is easy, because if $R$ is a field then $R$ itself ...
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1answer
22 views

$2^{p}\equiv1$ mod $2p+1$ for certain $p$

Let $p$ be a prime number such that $p\equiv3$ mod $4$ and $2p+1$ is also a prime number. It is well known that $2^{p}\equiv1$ mod $2p+1$, but I haven't been able to prove it.
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An integral domain and its field of fractions.

I'm trying to solve the following problem: Let $R$ be a integral domain which is not a field and $K$ its fractions field. Show that a non-zero module $R$-homomorphism from $K$ to $R$ does not ...
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2answers
48 views

Restriction of a group homomorphism to a normal subgroup

Suppose $f:G\to G$ is a group homomorphism and let $N\trianglelefteq G$. What can we say about $f$ if restriction of $f$ to $N$ is an identity on $N$? Can we say anything "nice" in this situation.
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$a,b$ are elements of the group $G$. Why $|ba|\leq |ab|$ in the following scenario?

A scenario: The order, $n$, of $ab$ (and hence, by definition, the order of the cyclic subgroup $\langle ab\rangle$) is finite (thus, the order of $ba$ is finite). Then $(ab)^n=e$. So ...
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1answer
35 views

Zeros of $317x^{2}-151xy+40y^{2}$ over $\mathbb{F}_{31}$

Let $K:=\mathbb{F}_{31}$ and $f(x,y):=317x^{2}-151xy+40y^{2}$. I have to find out if there exists any point $(a,b)\in K^{2}$ such that $f(a,b)=0$ and $a\neq0$ or $b\neq0$.
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2answers
33 views

$K \le B\le F$, If $F$ Galois over $K$ $\Rightarrow$ $F$ a Galois over $B$

Let $F$ a Galois over $K$, and let $B$ be a subfield of $F$ such that : $K \le B\le F$ $\Rightarrow$ $F$ a Galois over $B$ PROOF: $F$ is a splitting field of $f \in K[x]$ separable over $K$. ...
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1answer
35 views

Repetition of elements in a group, where am I wrong?

I am employing a particular method by which I am getting absurd results. I can not figure out where I am mistaking. Say $G$ is an abelian group of order 9. Let $a \in G$, so I am representing the ...
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2answers
76 views

The difference between vector space and group

When comparing the difference between the definition of vector space, I see that the main job is that vector space defines a scalar product while the group not, so here list two of my questions? ...
2
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2answers
75 views

Isn't $x^2+1 $ irreducible in $\mathbb Z$, then why is $\langle x^2+1 \rangle$ not a maximal ideal in $\mathbb Z[x]?$

Isn't $ x^2+1$ irreducible in $\mathbb Z$, then why is $\langle x^2+1 \rangle$ not a maximal ideal in $\mathbb Z[x]$? $ x^2+1$ cannot be broken down further non trivially in $\mathbb Z[x]$. hence, ...