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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How to solve the equation $x^2+Dy^2=\alpha$ over finite fields

It is known that the equation $x^2+Dy^2=1$ is solved over finite fields $\mathbb{F}_q$ and we can point out the solutions . I wonder can we give solutions for the equation $x^2+Dy^2=\alpha$ for any ...
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The Fifteen Puzzle and $S_n$ [duplicate]

I was studying permutation groups from the book "Abstract Algebra and Applications" by Karlheinz Spindler in which page 553 I came across the following interesting problem. It is on the famous "The ...
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1answer
38 views

Subgroups of Abelian Group of order 1000

Suppose you have an abelian group of size 1,000. How many subgroups does it have? I know there are 9 such groups from $1,000 = 2^3 \times 5^3$ giving us 3 of order $2^3 \times$ 3 of order $5^3$ ...
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40 views

Prove the polynomial is irreducible [duplicate]

I tried this problem for a while, but didn't see the application of Eisenstein's irreducibility criterion here. All the coefficients, including the leading coefficient, are equal to 1. p is a prime ...
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2answers
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A bit confused about definition of set of mappings in Herstein's Algebra

I have been just trying to start working with the books Topics in Algebra by I.N. Herstein, and I am having a bit of trouble understanding a definition. It is, Definition: If $S$ is a nonempty ...
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Prove that $x * y = \frac{x+y}{1+xy}$ is a stable part of $G=(-1, 1)$

I have to prove that the result of $x * y \in G$ so $\frac{x+y}{1+xy} \in (-1, 1)$. So $x > -1$ and $y > -1$ at the same time $x < 1$ and $y < 1$. If I multiply the first 2 expressions I ...
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Q-automorphisms determind by associates to id-element?

Let's say you consider the Galois group of $G(\mathbb{Q[\sqrt{3},\sqrt{2},i]}/\mathbb{Q})$. This is just an example. Is it correct that the $\mathbb{Q}$-automorphisms is determined up to associates? ...
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31 views

Relaxing the subgroup requirement for cosets

The definition I have for left cosets is as follows: Let $G$ be a group and let $H$ be a subgroup of $G$. A left coset of $H$ in $G$ is a set of the form $gH=\{gh:h\in H\}$ for some $g\in G$. ...
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1answer
43 views

Question from 14.6 “Galois Groups of Polynomials” from Dummit and Foote

I am confused in the proof of proposition 30 in Dummit and Foote on page 608. Near the end of this "proof" he goes on to say, By the Fundamental Theorem of Galois Theory, the fixed field of ...
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2answers
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Groups and Cent(). [closed]

Having some issues with this problem. I understand the concept that is required for a group as it must have the following four properties: $G$ is closed under the operation * The operation * is ...
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1answer
28 views

Induced homomorphism $\phi^*$ from $G/M \to H/N\ ?$

Let $\phi : G \to H/N$ be a homomorphism where $G$ and $H$ are groups and let $M \unlhd G$ and $N \unlhd H$. Now when does $\phi$ induces a homomorphism $\phi^*$ from $G/M \to H/N\ ?$ When $M ...
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23 views

Finding the size of a permutation group

Let A be an m by n matrix (you can assume that all elements of A are distinct). By a "legitimate transposition" on A, I mean an operation on A that swaps the (i,j)th and (k,l)th elements of A for some ...
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1answer
28 views

Separability implies flatness, in a special case

A nice theorem of Wang, Corollary 9 of A Jacobian criterion for separability, says the following: Let $B=A[z]=A[Z]/(h(Z))$. If $B$ is a separable algebra over $A$, then $B$ is a flat module over ...
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3answers
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What Is An Endomorphism Monoid?

I know why an object G gives you a group when you take all its automorphisms. But how does an object G give you a monoid when you take all its endomorphisms? What does it mean to compose two ...
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1answer
41 views

Ring localization and ideals

I'm trying to solve a couple of problems involving ring localization and I'm not sure if my solutions are right or if I understand the idea of localization correctly. Let $A$ be a commutative ...
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51 views

Pureness of Vector States

How does one show that irreducibility is equivalent to a vector state being pure? In what follows I will fill in the details of the question: Let $\mathcal{H}$ denote a Hilbert space and let ...
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1answer
66 views

A few tasks from group theory

Can you tell me, if my solutions are good? Mapping $f:\mathbb{Z}\rightarrow G$ with $f(k)=g^k$ is group homomorphism and Image(f) is abelian subgroup of G with $|\langle g \rangle |=ord(g)$ ...
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1answer
39 views

Descending chain of ideals becoming stationary

Exercise 3.7 of Algebraic Number Theory (Neukrich) is: In a noetherian ring R in which every prime ideal is maximal, each descending chian of ideals $\mathfrak{a_1 \supset a_2 \dots}$ becomes ...
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1answer
17 views

Content of Polynomials and Gauss's Lemma

I am getting stuck on a little part of a proof: Let $R$ be a PID and let $K=$Frac$(R)$. If $f\in R[x]$ and $f=gh$ with $g\in R[x]$ of content 1, show that $h\in R[x]$. We can clear the denominators ...
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54 views

how to show naturally isomorphic

I have a homological exam on Saturday , and I have some problem to understand of naturally isomorphic.my problem . the end of this theorem must proof naturally isomorphic $T_n $and ...
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38 views

How to describe $\#\{0\leq x<n:\gcd(x,n) \text{ is prime}\}$ the primes in $\mathbb{Z}/(n)$.

The above set actually comes from the following: In $\mathbb{Z}/(n)$ an ideal is prime if it is generated by an element $x$ such that for the integer representative $x$ we have $\gcd(x,n)=p$. To see ...
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35 views

Generalized fact in ring theory about irreducible elements

It is quite easy to show that for $A$ an integral domain, an element $a \in A$ is irreducible if and only if the principal ideal $\langle a \rangle$ is maximal for inclusion among proper principal ...
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42 views

Idempotent semiring

Let $R$ be a semiring. For $a\in R$,we define $t_a(x)=x+a$ for all $x\in R$. Prove that $R$ is idempotent(with +) and $1$ has an infinite order if and only if for all $a,x,y\in R$, ...
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15 views

Subalgebra of algebra of derivations

I would like to prove that a finitely generated subalgebra of the algebra of derivations of a finite algebra over the ring $F$ is finite over $F$. I have absolutely no idea how can I do it.
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32 views

Proving factorization into prime ideals in a Dedekind domain

Let $\mathcal O_K$ be a Dedekind domain and $\mathfrak p$ a prime ideal in $\mathcal O_K$. Assume that we have shown the existence of a fractional ideal $\mathfrak p^{-1}$ such that ...
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1answer
54 views

The existence of a polynomial factor

Given two polynomials $p_1(x_1,\dots, x_m)$ and $p_2(x_1,\dots, x_n)$ over reals, where $m > n$, and we know that $p_2(x_1,\dots, x_n)=0 \implies p_1(x_1,\dots, x_m) =0$. My question is: ...
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31 views

Galois group for the algebraic closure of a finite field

If $N$ is the algebraic closure of a finite field $F$, prove that $\operatorname{Gal}(N/F)$ is an abelian group and that any element of the Galois group has infinite order. If $N$ were a finite ...
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2answers
60 views

Abstract Algebra - Congruence Class Roots

How do we find the roots of ${x^3 + x + 1}$ in $ {Z_2[x]} $ The elements of the congruence class are: $$0, 1, x, x + 1, x^2, x^2 + 1, x^2 + x, x^2 + x + 1$$ as they have to be of the form $ax^2 + bx ...
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Divisibiltiy of the order of elements in a group

Let $G$ be a finite group and ket $y \in G$. How many elements $x \in G$ are there such that the order of $y$ is divisible by the order of $x$
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Let $\Phi : R \rightarrow R'$ be a ring homomorphism, where $R,R'$ are rings with unity. Then which of these is true?

Let $\Phi : R \rightarrow R'$ be a ring homomorphism, where $R,R'$ are rings with unity. Then which of these is true : $(i)~\Phi(1)=1 ~\forall~$ rings $R,R~'$ with unity $(ii)~\Phi(1) \ne 1 $ for ...
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1answer
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Is there a general formula for finding all subgroups of dihedral groups?

It seems that $\{e\}, \{e,s\}, \{e,rs\}, \{e,r^2s\},...,\{e,r^{n-1}s\}, \{e,r,r^2,...,r^{n-1}\}, D_n$ are always subgroups of $D_n$. Especially when $n$ is odd, these seem to be the only subgroups. ...
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1answer
32 views

A group of even order must contain an odd number of elements of order 2. [duplicate]

I tried it using proof by contradiction: Suppose that there are even number of elements of order $2$. Call them $x_1, ..., x_{2m}$, where $x_1,...,x_{2m} \neq e$. Then, consider the set $G=G ...
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To check whether $\langle 3+i\rangle$ is a maximal ideal or not in the ring of Gaussian integers $\Bbb Z[i]$

To check whether $\langle 3+i\rangle$ is a maximal ideal or not in the ring of Gaussian integers $\mathbb{Z}[i]$. Attempt: $\mathbb{Z}[i]/\langle 3+i\rangle$ is isomorphic to $10 \mathbb{Z}$ which is ...
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Field Question Proof with Axiom 4

Prove that if $(F,+,⋅,0,1)$ is a field, then there is no element $w ∈ F$ such that $0 \cdot w = 1$. Note that Axiom 4 from lecture (aka "M4" in the textbook) ensures that for $x ≠ 0$, there is a $w ∈ ...
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The multiplicative group of all complex $2^n$-th roots of unity, where $n = 0 , 1, 2 , \ldots$

Let $G$ be the multiplicative group of all complex $2^n$-th roots of unity, where $n = 0 , 1, 2 , \ldots$. Then assess the following claims: Every proper subgroup of $G$ is finite. $G$ ...
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Visual approach to abstract algebra

I'm currently finding abstract algebra to be very fascinating. However, one of the things that pulls me back is that I sometimes find it hard to understand something visually. For example, one could ...
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1answer
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In an SES of chain complexes in an abelian category two of complexes exact implies the third is exact.

Consider a short exact sequence of chain complexes: $$0_{\cdot} \rightarrow A_{\cdot} \xrightarrow{f} B_{\cdot} \xrightarrow{g} C_{\cdot} \rightarrow 0_{\cdot}$$ If any two of ...
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Question about the assumptions to have $G \simeq H\times K$

I've been looking this fact: Let $G$ be a group, with $G$ abelian. Let $H$, $K \leq G$, with $G=HK$ and $H\cap K=\{e\}$. Then, we have that $G \simeq H\times K$. And my question is: We know ...
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1answer
47 views

Principal ideal domain, $\forall x=(x_1,x_2)^t \in R^2~~\exists G \in SL_2(R) : Gx=(\gcd(x_1,x_2), 0)^t$

Let $R$ be a principal ideal domain. Prove that for every $x=(x_1,x_2)^t \in R^2$ exists a matrix $G \in SL_2(R)$ for which $Gx=(\gcd(x_1,x_2), 0)^t$. I think it's easy, but do not know how to ...
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Calculating global sections of sheaves

Consider the usual projective space $\mathbb{P}^{1} = \mathbb{C} \cup \{\infty\}$, and the Weil divisor $D = \{0\} \subset \mathbb{C} \subset \mathbb{P}^{1}$. Writing projective space as the union of ...
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Orbit, Stabiliser help please?

Let $Q$ denote the rectangle with the vertices $C_1=(2,1), C_2=(-2,1), C_3=(-2,-1)$ and $C_4=(2,-1)$. Describe the elements of the symmetry group $G$, of $Q$. Note that $G$ permutes the edges of ...
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Can We Always Build a Field out of an Integral Domain?

Link to Hungerford's Text Let $R$ be an integral domain, and $F$ its quotient field (or field of fractions). Assuming that $\phi: R \rightarrow F$ is isomorphic, $R[x]$ is isomorphic with $F[x]$ ...
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Does the ring of continuous functions determine $\mathbb R^n$?

I have two related questions which are just making the question asked in the title more specific: (a) Is every ring homomorphism (or maybe $\mathbb R$-algebra homorphism) between rings of the form ...
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1answer
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Injective map from integral domain to integrally closed domain? [closed]

If you have an injective map from an integral domain to an integrally closed domain, is that necessarily an integral extension? If so, is there an induced injective map on the respective field of ...
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Prove that $I_1^m\cap I_2^m \cap \dots \cap I_r^m=I_1^m\cdots I_r^m$, where $I_1,…,I_r$ are ideals in $k[x_1,…,x_n]$ and are comaximal.

This is an exercise from Ideals, Varieties and Algorithms by Cox, etc. If $I_i$ and $J_i=\cap_{j\ne i}I_j$ are comaximal for all $i$, where $I_1,...,I_r$ are ideals in $k[x_1,...,x_n]$, prove that ...
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176 views

Is the zero ring a domain?

Is the zero ring usually considered a domain or not? Wikipedia says: The zero ring is not an integral domain; this agrees with the fact that its zero ideal is not prime. Whether the zero ring ...
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Why the order of sylow 2 subgroup of SL$_2(F)$ where $F$ is a fielf of size $p^r$ is $n=$ord$_2(p^2-1)+$ord_2$(r)$?

I am trying to understand the sylow 2 subgroups of $SL_2(\mathbb F_{p^r})$ where $p$ is prime and $r\in \mathbb N$. In this wikipedea page, the para is ended by "... where ...
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21 views

Algebra. If $R$ is a commutative associative ring with neutral element.

Algebra. If $R$ is a commutative associative ring with neutral element. Then if R|P is an integral domain ($ab \in R|P \implies a=0 \ or \ b=0$) , then $1 \notin $ P ? This is the only thing I can ...
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1answer
47 views

Integral extension of local ring

I suppose this is a classical result, but I'm having problems to prove it. I want to prove that if $R$ is a commutative local ring and $R\subset S$ is an integral extension, then $S$ is also ...
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33 views

Confusion on a proposition of an algebra book

I have the following propostion from my algebra book: Prop 2.42. Let $f: X \to Y $ be a bijection. Suppose $A \subseteq X $ and $B \subseteq Y $. Assume that $f(A) \subseteq B $ and $f^{-1}(B) ...