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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Why in a graded ring $A$ finitely generated that's an algebra over a field $K$ every maximal ideal is a $K$-subspace?

Probably this question has already been asked, but I'm very bad in find old question and I searched for half an hour, so I'm asking it again. I suppose that's true beacuse my professor used this fact....
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0answers
26 views

Condition for an integer prime to be a Gaussian prime

I have a basic question: To show that an integer prime $p$ is a Gaussian prime (i.e. a prime in the ring of Gaussian integers $\mathbb Z[i]$) if and only if the equation $x^2+y^2=p$ has no integer ...
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3answers
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Prove that $Z(S_n) = \{(1)\}$ for every $n \geq 3$. Induction

I wonder if this questions can be done by induction. $S_3 = \{(1),(12),(13),(23),(123),(132)\}$ $Z(S_3)$ contains all the elements in $S_3$ that commutes with all the element in $S_3$ We can easily ...
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84 views

Show that a group with $p^k$ elements where $p$ is prime has a subgroup of order $p$

Proof- Pick an element $a \in G, \, a\not= e$ Now order $(a) = p^t$ for some $1 \leq t \leq k$ $\,$(by Lagrange) If I could show that $t \not= k$ so $G$ is not cyclic, I could use an inductive ...
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1answer
53 views

By Zorn's Lemma, a module can be finitely generated by its some elements and a submodule?

Take A as an R-module, and B is its submodule. We can get a submodule $B_1$ of A generated by B and an element $b_1 \in A\backslash$B. By the similar way we can get a submodule $B_2$ of A generated by ...
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1answer
136 views

Let $G$ be a group. Let $G/K$ be an abelian group. Prove that $C=\{e\}$.

I'm stuck at this exercise: Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Also suppose that $p \not\equiv 1$ (mod $r$), $p \not\...
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1answer
218 views

Classification of finitely generated multigraded modules over $K[x_1,\ldots,x_n]$?

Let $K$ be a field and $R=K[x_1,\ldots,x_n]=\bigoplus_{a\in\mathbb{N}^n}Kx^a$ the multigraded polynomial ring. Have finitely-generated multigraded $R$-modules been classified? Are they of the ...
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1answer
52 views

Submodule of a finitely generated module over a Dedekind domain

If $R$ is a Dedekind domain then is a submodule of a finitely generated $R$-module also finitely generated over $R$? Many thanks in advance.
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36 views

Homomorphism rings

$Hom(R,S)$denote the set of all homomorphisms of rings. $Char$ denote characteristic of a ring. 1) If either $Char R \neq 0$ but $Char S=0$ then $Hom(R,S)=\{0\}$ For example $Hom(\mathbb{Z}_{n},\...
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Is it true that $aH = bH$ iff $ab^{-1} \in H$

Let $H$ be a subgroup of a group $G$. The I know that for $a,b\in G$ we have $aH = bH$ if and only if $a^{-1}b \in H$. My question is if it is also true that $aH = bH$ if and only if $ab^{-1} \in H$? ...
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39 views

Prove that $\phi_g$ is an automorphism.

Let $G$ be a group, and $g$ be an element of $G$. The function (?) is defined as follows: $\phi_g:G\to G$, $\phi_g(h)=ghg^{-1}$, for every $h\in G$. Prove that $G$ is an automorphism. I am a ...
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3answers
99 views

direct product of cyclic and non-cyclic group together.

consider direct product of two finite groups, one is cyclic and the other one is not, is the direct product cyclic? if both groups are not cyclic,what we can say about direct product of them? I ...
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0answers
25 views

Example of a factorization into irreducible elements that is not unique? [duplicate]

I was reading about factorization into irreducible elements but could not think of any cases off the top of my head where the factorization would not be unique. Could you share some examples? Thanks!
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1answer
75 views

What is the Grothendieck group of finitely generated $R[G]$-modules?

Let $R$ be a ring with unity, $G$ a finite group and $R[G]$ the group ring. What is the definition of the Grothendieck group of finitely generated $R[G]$-modules? How is this connected to the ...
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1answer
57 views

a well defined map …

Consider a variety $V$ in $\mathbb{A}^n$, $I=I(V) \subset k[X_1, \cdots, X_n]=R$ ($k$ a field) and $P \in V$. We define the following : \begin{equation*} \mathcal{O}_P(\mathbb{A}^n) = \left\{ \left[ \...
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Example of algebraic structure that is non distributive for BOTH distributive laws and how to do computation in them?

(Apologies if this one sounds like I have not done much research, or I did not aware already have an answer, but I have been searching everywhere and all of these structures presented here, even ...
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203 views

To prove $\phi(mn)\phi(d)=\phi(m)\phi(n)d$ without explicitly computing the phi function values

If $m,n$ are positive integers with g.c.d.$(m,n)=d$ , then we can show by explicitly computing respective totients that $\phi(mn)\phi(d)=\phi(m)\phi(n)d$, I want to know, is there any more elegant way ...
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Direct products, direct sums and coproducts in category of groups

I have couple questions about terms I mentioned in the title. Why we don't define direct sum of non-abelian groups (subset of direct products which consists of elements with almost every component ...
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3answers
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Does there exist a group isomorphism from Z to ZxZ?

I know that $\mathbb Z$ and $\mathbb{Z}\times\mathbb{Z}$ have the same cardinality because you can create a bijection between the two. The example I was taught is Cantor's pairing function, which maps ...
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1answer
36 views

$M\simeq D$ and $N\simeq D$ then $M\oplus N\simeq D\oplus D$?

Suppose $D$ is a PID and $M,N$ are $D$-modules and we have $\varphi_1:D\longrightarrow M$ and $\varphi_2:D\longrightarrow N$ isomorphism. Then if I take $\varphi:D\oplus D\longrightarrow M\oplus N$, $...
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40 views

Comparison of abstract algebra structures

I made a comparison table of the different abstract algebra structures to help myself refresh with these concepts. Can some kind souls here help cross check the basic correctness? (Any other ...
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1answer
100 views

Is a prime principal ideal which is not maximal among principal ideals always idempotent?

Let $R$ be a commutative ring with identity, $P$ a prime principal ideal of $R$. Suppose that there exists a proper principal ideal $I$ of $R$ which is strictly larger than $P$ (i.e. $R\supsetneq I\...
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Group Isomorphism [closed]

Let $$H= \left\{\left( {\begin{array}{ccc} 1 & b \\ 0 & 1 \\ \end{array} } \right)\Bigg\vert b\in \mathbb{R}\right\}$$ be a group under multiplication. How do I show that $H\cong \...
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1answer
55 views

If $\gcd(|x|,|y|) = 1$ then $|xy| = \mathrm{lcm}(|x|,|y|)$ in an abelian group.

I am trying to prove this If $\gcd(|x|,|y|) = 1$ then $|xy| = \mathrm{lcm}(|x|,|y|)$ in an abelian group. My idea was we have the following $|x||y| = lcm(|x|,|y|)\times gcd(|x|,|y|)$ since we ...
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1answer
137 views

How do I prove that there exists a cyclic subgroup of order lcm of orders of cyclic subgrpups of an abelian group?

Before I start, please note that this post is not duplicate Let $G$ be an abelian group. Let $H,K$ be finite cyclic subgroups of $G$ such that $|H|=r,|K|=s$. Then, how do I prove that there exists ...
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1answer
81 views

I have to show this polynomial is irreducible. [closed]

Suppose that $p(x)=x^9+x^8+x^4+x^2+1 \in \mathbb{Z}_2[x]$. I have to show this polynomial is irreducible.
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1answer
32 views

Clarification on Homomorphism and Automorphism of Vector Spaces

I've been struggling to connect some of these concepts for a while now and seem to be confusing myself more than helping myself by continuing to think about them. Could someone confirm or deny my ...
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353 views

Groups involving matrices

Let $$H= \left\{\left( {\begin{array}{ccc} 1 & b \\ 0 & 1 \\ \end{array} } \right)\Bigg\vert b\in \mathbb{R}\right\}$$ How do I show that H is a group and under which operation? I ...
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1answer
44 views

Over a Bezout domain $R$, show every finitely generated submodule of $R^n$ is free

Let $R$ be an integral domain such that every finitely generated ideal of $R$ is principal. Show that every finitely generated submodule of $R^n$ is free. We know if $R$ is a PID, then the statement ...
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2answers
62 views

Describe the cosets of the subgroup $\langle 3\rangle$ of $\mathbb{Z}$

Describe the cosets of the subgroup $\langle 3\rangle$ of $\mathbb{Z}$ The problem I have is $\mathbb{Z}$ is infinite. So we know that $\langle 3\rangle=\{0,3,6,9,12,\ldots\}$ and I know the ...
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0answers
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Naively showing that $A_n$ mod a nontrivial normal subgroup is abelian.

Suppose $H \lhd A_n$ is a nontrivial normal subgroup of the alternating group on $n$ letters. Without using the fact that $A_n$ is simple, prove that $A_n/H$ is abelian. Can this be done? I will ...
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1answer
36 views

Does $K[x]/(p(x))$ contain the splitting field of $p(x)$?

Let $K$ be a field. And $p(x)$ be an irreducible polynomial in $K[x]$. Then, does $K[x]/(p(x))$ contain the splitting field of $p(x)$ ? If yes, can you give me the sketch of a proof ?
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polynomial associates if constant multiples

I have polynomials $2x^2+3x+1$ and $3x^2+2x+4$. I need to figure out if they are associates in $\mathbb{Q[x]}$ and $\mathbb{Z_5[x]}$. I know that two polynomials are associates iff they are constant ...
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1answer
84 views

$\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$?

I need to show that the extension $\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$, and compute its Galois group. I am learning Galois theory by myself and got stuck in this exercise. I ...
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0answers
24 views

Show that invertible elements of the algebraic closure of $F_p$ is not cyclic

I want to show that invertible elements of the algebraic closure of $F_p$ is not cyclic, where p is a prime. I know that the algebraic closure of $F_p$ is countably infinite, since it is equal to the ...
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44 views

Irreducible polynomial in $\mathbb{Z_2}$

I know that $x^2 + 1$ is irreducible over $\mathbb{R}$, since the $\sqrt{-1}$ is not in the reals. How can I verify whether or not $x^2 + 1$ is irreducible over $\mathbb{Z_2}$? So far I have ...
3
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1answer
122 views

The number of Sylow subgroups on $G$ with $|G|=pqr$

I'm doing a part of an exercise and I don't know how to go on. Here it goes: Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Show $G$ ...
5
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1answer
217 views

Let a~b iff there is an $x \in G$ such that $a=xbx^{-1}$

Question: Let G be a group. In each of the following, a relation on G is defined. Prove there is an equivalence relation. Let a~b iff there is an $x \in G$ such that $a=xbx^{-1}$ Here is my ...
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1answer
91 views

Bourbaki's proof of normal basis theorem Part 2

Let $K/k$ be a finite Galois extension of a field $k$, $G$ its Galois group. The normal basis theorem states as follows. There exists an element $\alpha$ of $K$ such that $\{\sigma(\alpha)\ |\ \...
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1answer
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Finding normal basis for GF(q^m) over GF(q)

Could you kindly explain, how can one find a normal basis for GF$(3^6)$ over the GF$(3^2)$? As I understood, I should start with finding the polynomial in a form $$a(x^2) + (a^9)x + a^{81},$$ which ...
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1answer
44 views

Minimal polynomial questions

I have to find the minimal polynomial for $cos(2\pi/5)$ and $sin(2\pi/5)$ like to know what is wrong with my attempt: $cos(4\pi/5)=cos(6\pi/5) = 4cos^3(2\pi/5)-3cos(2\pi/5)$, that gives $cos^3(2\...
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2answers
351 views

Difference between Stabilizer and Centralizer?

I know that the Centralizer of an element $a$ in a Group $G$ is defined as follows $$C_G(a) = \{ g \in G \space | \space ga = ag \}$$. It can also be defined as follows $$C_G(a) = \{ g \in G \space ...
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1answer
32 views

Inductive limit of A-algebras

I try to compute a pushout in the category of commutative $A$-algebras, where $A$ is a commutative ring with unity. My question is if there is some abstract nonsense which gives me a simple ...
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Advice on finding counterexamples

I am reaching out for specific advice on how one should go about finding counterexamples. It seems almost every time I've ever attempted a "find a counterexample" problem, I have to cheat by asking a ...
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1answer
58 views

Equivalence of Galois groups of two different splitting fields of the same polynomial

All fields are in $\mathbb{C}$ Let $f$ be a polynomial with coefficients in the field $F$. Let $F_1$ be a Galois extension of $F$ such that its Galois group $G(F_1/F)$ is cyclic and has prime order. ...
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Are these proofs all right? (Automorphism group of a string).

Background. Let $G$ be the automorphism group of a string $s$, ie. $G = \langle (i,j) : i \lt j, s[i] = s[j]\rangle$. Then $G$ is a normal subgroup of $S_{|s|}$ the symmetry group on $|s|$ symbols. ...
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82 views

all elements of ($Z$/p$Z$)* are cubes

Let $p$ be a prime An element $a \in$ ($Z$/p$Z$)* is called a cube if there exists $b \in$ ($Z$/p$Z$)* such that $a = b^3$ How to show that all elements of ($Z$/p$Z$)* are cubes ? And if $p \equiv ...
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Ideal of a commutative ring with $30$ elements.

Suppose that $R$ is a commutative ring and $|R|=30$. If $I$ is an ideal of R and $|I|=10.$ Prove that $I$ is a maximal ideal.
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1answer
191 views

$F$ is a field iff $F[x]$ is a Principal Ideal Domain

A commutative ring $F$ is a field iff $F[x]$ is a Principal Ideal Domain. I have done the part that if $F$ is a field then $F[x]$ is a PID using the division algorithm and contradicting the ...
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2answers
100 views

Group Theory: Showing that a subgroup is isomorphic to a product of groups

I have the following question, where the topic being tested is cosets, order and Lagrange's theorem: Suppose that every element $x$ in a group $G$ satisfies $x^2 = e$. Prove that $G $is abelian. ...