Questions tagged [abstract-algebra]

For questions about monoids, groups, rings, modules, fields, vector spaces, algebras over fields, various types of lattices, and other such algebraic objects. Associate with related tags like [group-theory], [ring-theory], [modules], etc as necessary. To clarify which topic of abstract algebra is most related to your question and help other users when searching.

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Proof the follow exercise of group theory [closed]

let $a \in G$ where $G$ is finite. If $a^n$ has $m$ conjugates and $a$ has $k$ conjugates, the m|k.
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How do i generate cubic coordinate-space traversal functions?

Goal- generation of a function f(x) such that for each x = 0 to infinite, f(x) outputs a point in a coordinate space ...
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Show that if $G$ is finite and $G$ acts on $X$ and it is transitive, then $1=\dfrac{1}{|G|}\sum_{g\in G}|\operatorname{Fix}(g)|$ [closed]

I know you have to relate the $\sum_{g\in G} |\operatorname{Fix}(g)|$ with $\sum_{x\in X} |\operatorname{Stab}(g)|$ and use the orbit theorem and stabilizer but I get very confused.
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Extension of a representation

My question is related with this Representation abelian subgroup of an abelian group. I want to show that X extends to a representation of G, which I think it's obvious from G being defined by its ...
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-4 votes
0 answers
63 views

proof there exist element $1_{G} \neq x \in G$ such that $C(x)=G$.

let G be a finite group and $A\triangleleft G$ such that $\left|A\right|\leq\left|G\right|^{1/2}$. proof there exist $1_{G}\neq x\in G$ such that $C(x)=G$. from the previous exercise , $\left|AA^{g}\...
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0 votes
1 answer
17 views

A link between multiplicative function on endomorphisms and their determinants

Take $E$ a $K$-vector space with $K$ a field with characteristic $0$. Then take $$ F:\operatorname{End}_K(E) \to K $$ a map such that $$F(1)=1$$ and $$ F(\phi)F(\psi) = F(\phi)F(\psi) $$ Then there ...
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1 answer
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The degree of simple radical extensions

Let $m \le n$ be positive integers. Does there necessarily exist a field extension $K/F$ such that $[K:F] = m$ and $K = F(u)$ for some $u \in K$ satisfying $u^{n} \in F$? In other words, given $m \le ...
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1 vote
0 answers
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What is the polynomial ring of $1$ over the cyclic group of $2$.

I'm talking about $\mathbb{F}_2[1]$ I think the answer would be $\mathbb{N}$ because each coefficient is $0$ or $1$, and since the first coefficient can only be $0$, we have the first element of $\...
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-4 votes
1 answer
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Can an algebraic field be redefined and an axiom added?

Can an algebraic field be redefined to allow division by 0 and an axiom included to define x/0 as x(1/0)? I was thinking of making a field, let's call it J for now, where numbers are written as a + bi ...
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1 vote
1 answer
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Proving that $\alpha\gamma$ and $\beta\gamma$ have no gcd where $\alpha=3$ , $\beta=(1+2\sqrt{-5})$ and $\gamma=7(1+2\sqrt{-5})$in ring $Z[√(-5)]$

I have the following question before me : $\alpha=3$ and $\beta=1+2\sqrt{-5}$ are two elements of ring $R=Z[\sqrt{-5}]$. $\gamma=7(1+2\sqrt{-5})$ is another element of this ring. I have to prove that ...
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1 vote
1 answer
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When the tensor product by a $K$-algebra is a faithful functor on the category of $K$-modules?

This is a quite general question, I know. Given a commutative ring $K$ and an algebra $K\to A$, there's an endofunctor $A\otimes_K-\colon K\text{-}\mathrm{Mod}\to K\text{-}\mathrm{Mod}$. I was ...
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Completion of a finite lenght module

I am reading a proof on Bruns and Herzog's book about Cohen-Macaulay rings and they assume the following propertie: If $(R,\mathfrak{m},k)$ is a Noetherian local ring and $M$ is a finite lenght module,...
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For every $(e_1,…,e_n)$∈{−1,1}n, there is an automorphism $\phi$ of $Q(\sqrt{p1},…,\sqrt{pn})$ with $\phi(\sqrt{p_i})=e_i\sqrt{p_i}$, 1≤i≤n

Let $En=Q(\sqrt{p1},…,\sqrt{pn})$ ,I don't understand how to proof "For every $(e_1,…,e_n)$∈{−1,1}n, there is an automorphism $\phi$ of En with $\phi(\sqrt{p_i})=e_i\sqrt{p_i}$, 1≤i≤n"....
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Is spliting field of $f(x) \in F[x]$ with $F$ a finite field also a finite field?

Let $F$ be a finite field, and $f(x) \in F[x]$ is some non constant polynomial, Let $E$ be the splitting field of $f(x)$ over $F$ , is $E$ always a finite field? My attempt let $\deg(f) = n$ then ...
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Let $E$ be the spliting field of $x^{p^n} - x$ over $\Bbb{Z}_p$ then $E$ has charateristic $p$

Prove if $E$ be the spliting field of $x^{p^n} - x$ over $\Bbb{Z}_p$ then $E$ has charateristic $p$. where $p$ is some prime number. My attempt: First, assume $E$ be a finite field, therefore the ...
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2 votes
1 answer
112 views

What is the general number of solutions of the equation $y^2=x^3-x\pmod{p}$?

What is the general number of solutions of the equation $y^2=x^3-x\pmod{p}$, where $p$ is a prime number and $p>3$?. Calculations suggest that the number of solutions to this equation is $p$ if $p\...
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0 votes
1 answer
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Is $\mathbb{Z}[\sqrt[3]{2}]$ a UFD? [duplicate]

Is the quotient ring $\mathbb{Z}[X]/(X^3-2)$ a UFD? I don’t have any idea to find whether some ring is a UFD or not.
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Math Questions Basic level [closed]

Find number of integers n such that n+3 divides n^3-3 ?
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1 answer
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How to prove that Quaternion's algebra over isomorphic to Mat2(Z [duplicate]

My ideas: I tried to build an explicit isomorphism, but as I think it is only possible when p = 1 (mod 4), and for p = 1 (mod 4) it get it. In my second attempt, I tried to look at them as vector ...
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0 answers
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If the product topology on $\mathbb{R}^2$ is all sets of the form $(a,b)\times(c,d)\subseteq\mathbb{R}^2$ what is product topology on $\mathbb{R}^4$?

Probably the answer to this question is too obvious for it to be asked but I want to make sure my assumption is correct. Consider the real line $\mathbb{R}$ with the topology that has basis consisting ...
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If Sigma(xn) and Sigma(yn) are convergent, show that Sigma(xn +yn) is convergent.

this is from exercise 3.7 introduction to real analysis bartle fourth edition number 4 i already tried to proof the Xn+Yn as a convergent but somehow it doesnt turn to proofed and still got some stuck
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Naming convention: Lie groups with finitely generated discrete part

Consider a Lie group $G$ and let $G_0$ be the connected component of the identity. Then $G_{\mathrm{dis.}}:=G/G_0$ is a discrete group. We are writing a (physics) article in which finite-dim. Lie ...
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2 votes
0 answers
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Representation abelian subgroup of an abelian group

Let G a non abelian group of order $p^{3}$ and exponent $p$, where $p$ is a prime. Let $w$ be a primitive complex p-th root of unity. So far, I have proved that: $Z(G)=[G:G]$ $|Z(G)|=p$ $G\cong \...
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0 votes
0 answers
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Bezout Lemma in a PID [duplicate]

The Bezout Lemma in the integers states that For any $a, b \in\mathbb Z$, let $g = \gcd(a, b)$, There exists $x, y$ such that $ax+by = g$. This can be generalized to a commutative ring that is a ...
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1 vote
2 answers
64 views

Finding specific isomorphism $\mathrm{Hom}(M, R)\otimes_R F \rightarrow \mathrm{Hom}(M, F)$

Let $R$ be a commutative ring with $1_R$. Let $M$ be a $R$-module and $F$ be a free $R$-module. How do I find a specific isomorphism to show that $$\mathrm{Hom}(M, R)\otimes_R F \cong \mathrm{Hom}(M, ...
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1 vote
1 answer
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A question in corollary of Hilbert Nullstellansatz

This corollary was part of my lecture notes in commutative algebra and I am having questions in proof of it. Statement: Let I be an ideal in $K[x_1,...,x_n]$ , K is algebraically closed . Then $ I (...
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-1 votes
1 answer
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Question in proof of a corollary of Hilbert Nullstellansatz

I have been reading commutative algebra from lecture notes and I have some questions in a proof of a corollary of Hilbert Nullstellansatz. Let R be a finitely generated k-algebra. Then for an ideal $...
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-3 votes
1 answer
34 views

Is C3 a subgroup of every group with index greater than 3 [closed]

Is $(e, a, a^{-1})$ a subgroup of every group? If not, what is the reason?
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9 votes
0 answers
131 views

$A,B$ such that $A\cap B=\emptyset$ and $A\cup B=\mathbb{R}$ and $B=\{x+y : x,y\in A\}$?

If set $A,B$ satisfy $A\cap B=\emptyset,A\cup B=I$, and $B=\{x+y : x,y\in A\}$, can $I$ be real number set $\mathbb{R}$? I think the answer is yes, but I can't construct it. If $A$ is odd number set, ...
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0 answers
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How to conclude the order of $G$ is $p^{m+n}$.

I have this exact sequence $$ 0\to \mathbb{Z}_{p^m}\overset{f}\to G\overset{g}\to \mathbb{Z}_{p^n}{\to} 0 $$ where $G$ is finitely generated. I want to prove that $|G|=p^{m+n}$ and a friend told me to ...
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Prove for the ideals $I,J$ that $J\nsubseteq I$

consider the ideals $I=\langle f_1,f_2\rangle$ and $J=\langle h_1,h_2\rangle$ in $\mathbb{Q}[x,y]$. I want to prove that $J\nsubseteq I$. I'm trying to do this by proving that every element from $J$ ...
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3 votes
0 answers
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Divisors in GCD domain [duplicate]

Given a GCD domain $D$: $D$ is a GCD domain if for every two elements $a,b\in D$, $\exists d \in D\setminus \{0_D\} $ such that: (GCD.I) $d\mid a$ and $d\mid b$ (GCD.II) If $\exists c\in D\...
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2 votes
0 answers
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Prove a nonempty set $G$ with an associative operation $\ast$ is a group iff the following equations are satisfied $y \ast g = h$ and $g \ast x = h$ [duplicate]

I'm currently working on an exercise and the body of the text for the exercise is as follows. I have a first draft of the proof but am missing some things and am unsure about some things as well so ...
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1 vote
0 answers
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Equivalent Definition of injective $R$-module using exact sequences [duplicate]

We call an $R$-module $D$ injective if one of the two equivalent conditions holds: If $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ is an exact sequence of $R$-modules then the induced ...
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2 votes
0 answers
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Prove that a normal subgroup $G$ of $S_4$ with $(12)\in S_4$ is equivalent to the entire group $S_4$ [duplicate]

Consider a normal subgroup $G$ of $S_4$. The simple transposition $(12)\in G$. Prove that $G=S_4$. I have already proved that the above case implies that $(12),(23),(34)\in G$. These are all the ...
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-3 votes
0 answers
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Questions in structure theorems for artinian rings [duplicate]

This theorem was covered in the lecture notes of commutative algebra from where I am studying and I am struck on it on some points. Statement: Every artinian ring is a finite product of artinian ...
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$A \subset B$ rings, if $\alpha \in B$ integral, then $A[\alpha]$ integral over $A$?

Notation is from Lang, beginning of Chapter VII. (Extension of Rings). Rings are commutative with identity. $\alpha\in B$ is integral over $A$ if subring $A[\alpha]$ is finitely generated $A$-module (...
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1 answer
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Showing that $x^3 - t$ is irreducible over $\mathbb{F}_3(t)$

I was reading the post Is $\mathbb{F}_3(t,t^{1/3})/\mathbb{F}_3(t)$ a normal extension? Is it separable? I do not understand, why we can use Eisenstein's criterion to show that $x^3 - t$ is ...
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1 vote
0 answers
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Uniqueness of the ternary Golay codes

In [Van Lint - Introduction to Coding Theory] the uniqueness of the binary Golay codes is shown quite easily. In essence, the proof boils down to the fact that there is only one 2-$(11,5,2)$-design up ...
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-1 votes
1 answer
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A is noetherian and every prime ideal of A is maximal then...

This proof was part of my lecture notes in Commutative algebra and I am having trouble following it. So, I am asking the question here. Statement: If A is noetherian and every prime ideal of A is ...
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-2 votes
1 answer
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Question in the proof that any Artinian ring is noetherian

This theorem is from my lecture notes of Commutative Algebra and I am struck on 2 points of the proof. Statement: Any artinian ring is noetherian. Proof: Let A be an artinian ring. Let $M_1 ,...,...
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1 vote
1 answer
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$M$ is linearly dependent if a set $N$ exists, such that $N\neq M$, $N\subset M$ and $Span(M)=Span(N)$.

V is the Vector space defined over $\mathbb{R}$. Let M be a subset of V. Prove that M is linearly dependent if a set N exists, such that $M\neq N$, $N\subset M$ and $Span(M)=Span(N)$. Let $0_v\neq x\...
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0 answers
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Where I am wrong in mod $p$ irreducibility test?

Mod p irreducibility test : Let $p$ be a prime an suppose that $f(x) \in \mathbb Z[x]$ with $\deg f(x) \geq 1$. Let $f_1(x)$ be the polynomial in $\mathbb Z_p[x]$ obtained from $f(x)$ by reducing all ...
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1 vote
0 answers
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Coinvariants of coaction $a\otimes b \mapsto \sum{\sigma_i(a)}\otimes \sigma_i(b)\otimes \sigma_i^*$

I've been studying Hopf-Galois Theory and currently I'm trying to understand some examples by writing all the explanations step by step by myself. The example I'm interested now is the classical ...
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-3 votes
2 answers
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Factoring polynomials modulo 3

Let $f(x) = x^5 + 2x^2 + 2x + 2 \in\mathbb Z_3[x]$. Then the irreducible factorization of $f(x)$ is $(x^2 +1)(x^3+2x+2)$ even though it does not have a root in $\mathbb Z_3$. How did we find that ...
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-1 votes
0 answers
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Is $S^{-1}(q_i)$ is a primary ideal in $S^{-1} A$?

This question was left as an exercise in class of commutative algebra and I am struck on it. Let A be a noetherian ring and $I=\cap_{i=1}^n q_i$ be a minimal primary decomposition. Let $p_i = \sqrt{...
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0 answers
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show that $x^3 = (1234)$ in $S_7$ has three solutions (and find them?) [duplicate]

As title. This is question 52 in Chapter 5 of Gallian’s Abstract Algebra, 10th edition. My current line of logic is as follows. We have $x^3 = (1234)$, which gives us $|x^3| = 4$, implying that $|x| = ...
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2 votes
1 answer
78 views

Maximal algebraic independent in commutative algebra over field

In field extension, maximal algebraic independent elements in a set of generators (generate by means of fraction of generators) will also be maximal alebraic independent amoung all subsets. It is then ...
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0 answers
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find all the prime ideals of $\mathbb{Z}_8$ [duplicate]

find all the prime ideal of $\mathbb{Z}_8$? My attempt : Positive divisor of $8$ are $1, 2, 4$ and $8$ So the ideal in $\mathbb{Z}_8$ are $(1)=\mathbb{Z}_8$ $(2)=\{0,2,4,6\}$ $(4)=\{0,4\}$ $(...
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4 votes
1 answer
282 views

Can mathematics distinguish left and right?

Imagine, a mathematician from another galaxy lands on the earth. Is there a way we can explain to him what is "counterclockwise" without showing him a picture? Things like Green's formula, ...
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