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Questions tagged [abstract-algebra]

Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, and modules, among other topics. Whenever you use this tag, please also include related tags like group-theory, ring-theory, modules, etc., in order to clarify which topic of abstract algebra is most related to your question and also to help other users find similar questions through the search engine.

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1answer
34 views

Kummer ring - special monic polynomial with zero at root of unity

Let $ \zeta_n = e^{2 \pi i / n} $ be the n-th root of unity. Let $$ P(z) = \sum_{k = 0}^{n-1} s_k z^k $$ be a monic polynomial over $ z \in \mathbb{C} $, specified by integer coefficients $ s_k \...
5
votes
3answers
160 views

Does there exist a group that is both a free product and a direct product of nontrivial groups?

Do there exist such nontrivial groups $A$, $B$, $C$ and $D$, such that $A \times B \cong C \ast D$? I failed to construct any examples, so I decided to try to prove they do not exist by contradiction....
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2answers
35 views

Determine whether the following relation is an equivalence relation

Determine whether the following relation B on A is reflexive, symmetric, and transitive. $$A=\{1,2,3\}$$ $$B = \{(1,1),(1,3),(3,2)\}$$ These are my guesses, because the textbook only showed an ...
2
votes
2answers
41 views

Example of a Galois extension $L/\mathbb{Q}$ with $\text{Gal}(L/\mathbb{Q})\cong\mathbb{Z}_{4}$

I propose that $L=\mathbb{Q}(i\sqrt[4]{2})$. Obviously $\mathbb{Q}(i\sqrt[4]{2})$ is the splitting field of $f=t^{4}-2\in\mathbb{Q}[t]$, since $N:=\{\text{zeros of $f$}\}=\{\sqrt[4]{2},-\sqrt[4]{2},...
3
votes
0answers
31 views

Associated Graded Algebra

I'm trying to work through Exercise III.27 of Lang's Algebra: Let $A$ be a filtered algebra, $A=\bigcup_{j\geq 0}A_{j}$. For $j\geq 0$, define $R_{j}=A_{j}/A_{j-1}$, with $A_{-1}=\{0\}$. Let $R=\...
0
votes
1answer
17 views

How to determine the kernel $\ker \varepsilon_{\sqrt 2}$ and the image $\varepsilon_{\sqrt 2}(\mathbb Q[X])?$

Consider the $\mathbb Q$-algebra homomorphism $\varepsilon_{\sqrt 2}:\mathbb Q[X]\rightarrow \mathbb C$ defined by $\varepsilon(X)=\sqrt 2$. How to determine the kernel $\ker \varepsilon_{\sqrt 2}$ ...
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1answer
46 views

When does a power series converge to a rational function?

Are there any results to determine whether the given power series of real variable converges to a rational function? I mean just analyzing the coefficients of the series. One way is to find the sum ...
1
vote
1answer
66 views

How to show that there are infinitely many prime numbers p such that the polynomial f has a zero in Zp? [duplicate]

Let $f\in \mathbb Z[X]$ be a polynomial of positive degree.How to show that there are infinitely many prime numbers $p$ such that the polynomial $f$ has a zero in $\mathbb Z/p \mathbb Z$ ? I have no ...
0
votes
0answers
13 views

Distinct elements in the quotient

How can find the elements in the Ring $\mathbb{Z}_5[x]/\langle (x+1)\cdot(x+3)\rangle$? if it is just $\mathbb{Z}/5\mathbb{Z}$, then it easy to find the elements are $\overline{0},\overline{1},\...
3
votes
1answer
66 views

Is the Pisot Triangle series known?

The Kepler triangle is built with powers of $\sqrt\phi$ to make a right triangle. The supergolden ratio can make a 120° triangle. It turns out that most Pisot numbers (Mathworld, Wilkipedia) 1 to 4 ($\...
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vote
1answer
35 views

Factor Theorem for Multivariate Polynomials [duplicate]

I am looking for neat ways of proving the following theorem: Let $F$ be a field and let $f \in F[t_1, ..., t_n]$ be a polynomial. If $f(\pmb{u}) = 0$ for some $\pmb{u} \in F^n$, then $f$ lies in ...
0
votes
1answer
23 views

The multiplicity of a root $r$ of a irreducible polynomial is a power of $p$ characteristic

$f$ is an irreducible polynomial over a field $K$ of characteristic $p$. $F$ is a splitting field of $f$ over $K$ and $u_1$ a root of $f$. I have shown that $f=[(x-u_1)\cdots (x-u_n)]^{[K(u_1):K]_s}$ ...
0
votes
1answer
36 views

How to find the generator of the following ideals?

How to find the generator of the following ideals $\cal a=${$F\in \mathbb Q[X]:F(i)=0$} in $\mathbb Q[x]$, $\cal b=${$F\in \mathbb Q[X]:F(\sqrt 2i)=0$} in $\mathbb Q[x]$? $\cal c=${$F\...
3
votes
1answer
30 views

The “span” of a $\mathbb{Z}$-module.

Consider the $\mathbb{Z}$-module $M=\mathbb{Z}^2$, $$ M_1=\{(a_1,a_2):2a_1+3a_2=0\} \text{ and } M_2=\{(a_1,a_2):a_1+a_2=0\}$$ Prove that $M$ is the internal direct sum of $M_1$ and $M_2$. To prove ...
3
votes
2answers
29 views

The number of ring homomorphism from $\mathbb Z[x,y]\to \mathbb{F}_2[x]/(1+x+x^2+x^3)$

Find the number of ring homomorphism from $\mathbb Z[x,y]\to \mathbb{F}_2[x]/(1+x+x^2+x^3)$. My attempt: the ring $Z[x,y]$ has three generators $1,x \ and\ y$ we want $1$ to map to $1.$ Since the ...
0
votes
1answer
52 views

For an integral domain $R$, the rings $R\times R $ and $R\times R\times R$ are not isomorphic

For an integral domain $R$, the rings $R\times R $ and $R\times R\times R$ are not isomorphic My attempt: On contrary suppose that both are isomorphic then if G is prime ideal of one ring then its ...
4
votes
3answers
53 views

A group has odd number of elements iff each element is a square, $a=b^2$.

Let $G$ be a finite group having order $n$, prove that $n$ is odd if and only if for each $a\in G$, there is $b\in G$ such that $a=b^2$. I first assume $n$ is odd, then let $a$ be an element in $G$, ...
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0answers
34 views

Why is clifford group a group?

Let $C(Q)$ denote the clifford algebra of vector space $Q$ with respect to a quadratic form $q:V \rightarrow \Bbb R$. Hence we have the relation $w^2 = Q(w) \cdot 1$ for $w \in V$. Let $\alpha:C(Q) \...
2
votes
1answer
38 views

Prove a subring of $R=\mathbb{Q}[i]$ is equal to $R$ itself or $\mathbb{Q}$

Consider the ring $R = \mathbb{Q}[i] = \{a + bi \mid a, b ∈ \mathbb{Q}\}$, the subring of $\mathbb{C}$ of all complex numbers with rational real and imaginary parts. Let $T \subset R$ be a ...
3
votes
1answer
54 views

Show for any simple ring without identity, R, that R is a division ring.

Question: Show for any simple ring without identity, R, that R is a division ring. My thought process is to consider an ideal generated by one element $r$, so I want to consider the ideal $rR$ = $ \...
1
vote
1answer
29 views

The localization of an ideal is equal to the localization of the ring

Suppose $m\subset R$ is a maximal ideal. Suppose $I\subset R$ is an ideal. I'm trying to understand these claims: If $m$ does not contain $I$, then $I_m=R_m$ as localizations of $R$-modules. If $m$ ...
3
votes
2answers
52 views

Find all of homomorphisms $Φ$ from $\mathbb C$[$x$]/$I$ to $\mathbb C$ that satisfies specific condition

I want to know how can I find ALL homomorphism that satisfies the condition mentioned in the question above. Please give me a method or answer in detail.
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0answers
21 views

What is a ring algebra? [duplicate]

Let $R$ be a ring. What does it mean to be an $R$-algebra? I missed the definition in my lectures and when I search online, I can only find definitions of what a ring is.
0
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0answers
29 views

What is the centralizer of an element of order 2 of the icosahedral group

The class equation of the icosahedral group, I, is $60 = 1 + 20+12+12+15$. The conjugacy class of order 15 contains elements of order 2, and their centralizers are of order 4. The centralizer contains ...
2
votes
2answers
39 views

Determine the class equation of the tetrahedral group

This is a question from Artin's algebra textbook. The tetrahedral group of rotations has 1 element of order 1, 8 elements of order 3 (rotations of $120^°$ around a vertex), and 3 elements of order 2 ...
2
votes
1answer
37 views

Is it true “If $X \subset Y$ then $\Bbb I(Y) \subset \Bbb I(X)$”(proper inclusion)?

Is it true "If $X \subset Y$ then $\Bbb I(Y) \subset \Bbb I(X)$; here I am using proper inclusion. Couldn't prove it though. Trying for long time please help. Actually I saw here "https://people.maths....
3
votes
0answers
63 views

Definitions of Group Cohomology

I am trying to understand group cohomology, and I have a very basic question. So as I understand it, let $\Gamma$ be a group, and $V$ be a $\Gamma$-module (which is essentially another abelian group ...
1
vote
1answer
29 views

How does the Frobenius elements play a crucial role in Galois representation?

How does the Frobenius elements play a crucial role in Galois representation? As far as I have understood that Frobenius endomorphisms specially Frobenius automorphism is a generator of a Galois ...
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vote
1answer
41 views

Rational expression $K(t)$ is a transcendental extension of $K$?

I know this question was asked before, but none of the previous threads end up answering the question satisfactorily enough for me. So let me try to summarize my problems succinctly: The notation $\...
2
votes
1answer
60 views

Let $\alpha$ denote the image of $x$ in $\mathbb Q[x]/(x^3 + 3x + 3)$

Let $\alpha$ denote the image of $x$ in $\mathbb Q[x]/(x^3 + 3x + 3)$. Express each of $1/α,1/(1+ \alpha),1/(1+ \alpha^2)$ in the form $c_2\alpha^2+c_1\alpha+c_0$ with $c_0, c_1,c_2 \in \mathbb Q$. I ...
1
vote
1answer
36 views

closed embedding is open iff ideal sheaf idempotent

Let $f: \mathrm{Spec}(R) \to \mathrm{Spec}(A)$ be a closed embedding of affine Noetherian schemes, given by the ideal $I = \mathrm{ker}(A \twoheadrightarrow R)$. If $I = I^2$, then it's not too hard ...
0
votes
2answers
63 views

Irreducible polynomial in $\mathbb C[x_1,x_2]$ also irreducible in $\mathbb C[x_1,x_2,…x_r]$? [duplicate]

Let $f_1(x_1), f_2(x_2)$ be polynomials in a single variable, of relatively prime degree, with complex coefficients. If $f_1(x_1)+f_2(x_2)$ is irreducible in $\mathbb C[x_1,x_2]$, then is it ...
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vote
0answers
49 views

Find all integers n, so that $n^2$ divide $2^n+1$ [duplicate]

Bonjour à tous! As the title suggest I want to find all the $n>1$ such that $n^2| 2^n+1$. My attempt: Let $p$ an odd prime dividing $n$. We know that $2^p+1|2^n+1$ $\textit{ (Isn’t it the only ...
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votes
2answers
58 views

Idempotents in $ \mathbb{Z}_n $ [on hold]

Let $ n=cd $ where $ c $ and $ d $ are co-primes. Then there are integers $ x $ and $ y $ such that $ xc+dy=1 $. How can it be proved that $ xa $ is idempotent in $ \mathbb{Z}_n $? Is converse ...
2
votes
1answer
44 views

$\mathbb{Z}$-module $\prod\limits_{\text{p prime} } \mathbb{Z}/p\mathbb{Z}$

I am trying to establish if the $\mathbb{Z}$-module $\displaystyle\prod_{\text{p prime} } \mathbb{Z}/p\mathbb{Z}$ is torsion-free. So I think that the elements of $\displaystyle\prod_{\text{p prime} }...
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0answers
31 views

$\mathbb{Q}/\mathbb{Z}$ is a torsion $\mathbb{Z}$-module

I'm reading through some chapters of Dummit and Foote Abstract Algebra book and in one of the examples from Projective Modules section they state that $\mathbb{Q}/\mathbb{Z}$ is a torsion $\mathbb{Z}$-...
2
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1answer
70 views

Prove a specific skew field K is commutative

Let $\mathcal{K}$ be a skew field such that there exists $p\geq 2$ prime with $$\underbrace{1+1+\ldots+1}_{p\text{ times}}=0$$ and for any $x\in\mathcal{K}$ there is a positive integer $n=n(x)$ so ...
2
votes
2answers
122 views

Is $\varphi(ab)=\varphi(b)\varphi(a)$ an equally valid definition of homomorphism?

Given the groups $G$ and $\bar G$, usually a homomorphism is defined as a map $\varphi \colon G \longrightarrow \bar G$ such that $\varphi(ab)=\varphi(a)\varphi(b)$. Now, if we'd define homomorphism a ...
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0answers
19 views

Associative enveloping algebra for non Lie algebras?

Is it possible for any algebra (for example commutative) to construct an associative enveloping algebra as we do for Lie algebras? If so, can I get any reference?
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0answers
22 views

Ideals in $R= \mathbb Z[\sqrt{-5}]/(3) $

I know that there are 9 elements in R, namely $a+b\sqrt{-5}$ with $a,b\in {0,1,2}$. Now I'm trying to find nontrivial ideals of R, and my book tells me that $(1+\sqrt{-5})=\{0,1+\sqrt{-5},2+2\sqrt{-5}\...
2
votes
0answers
21 views

Existence of infinitely many maximal ideals in …

Let $\mathscr{A}_p=\{f\in C[0,1]:f(p)=0\}$. Then we all know that every maximal ideal of $C[0,1]$ is of the form $\mathscr{A}_p$ for some $p\in [0,1]$. Instead of considering compact set $[0,1]$, if ...
0
votes
1answer
34 views

Find elements in quotient ring which satisfies specific condition [on hold]

Let $\mathbb { R } [ x ]$ be the polynomial ring in one variable over $\mathbb { R }$ . Let $I$ be the ideal of $\mathbb { R } [ x ]$ generated by the polynomial $x ^ { 3 } - 8 .$ Consider the ...
3
votes
4answers
87 views

Questions about the proof of (a $\star$ b)$^{−1}$ = b$^{−1}$ $\star$ a$^{−1}$

The book proves (a $\star$ b)$^{−1}$ = b$^{−1}$ $\star$ a$^{−1}$, where $\star$ is considered a binary group operation. I will state the book's proof and then follow up with my questions. Book's ...
1
vote
2answers
52 views

Why is $(a+b)^p = a^p+b^p$, where $a,b \in R$, a commutative ring with prime characteristic $p$?

Here is the answer from lecture notes. $(a+b)^p = \sum {{p}\choose{k}}a^kb^{p-k}$ and all the terms divide $p$ except $a^p$ and $b^p$ terms. So reducing (mod p) all terms are zero except the ones ...
1
vote
2answers
29 views

GCDs for the polynomial ring over a Galois field.

You can find many examples of computing the inverse of an element inside a Galois field. (For example here) What happens if we look at the polynomial ring over a Galois field and would like to ...
1
vote
1answer
41 views

Irreducibility of the following Polynomial over $\mathbb{Q}$

Take the polynomial $x^4 + 10x^2 + 1$. Is this irreducible over $\mathbb{Q}$? If so, what is the best way to show this? I know it can be rewritten: $$ (x^2 + 5)^2 -24 $$ Which can be simplified over ...
0
votes
1answer
14 views

Proving that two subsets of a set of bijections are either equal or disjoint

$X$ is a set and $X!$ is the set of bijections from $X$ to $X$. Let $A(m)$ be the set defined as $A(m)=\{f^x(m)\mid x\in\mathbb Z\}$ for some $f$ in $X!$ and some $m$ in $X$. Now, how do I prove that ...
0
votes
0answers
66 views

Pythagorean closure

Im reading the book "Galois Theory" by Ian Stewart $(4$th Edition$)$. Here the author defines the Pythagorean closure as follows: Definition:The Pythagorean closure $\mathbb{Q}^{PY}$ of $\mathbb{Q}...
0
votes
2answers
42 views

Is every ideal in $K[x, y]$ of finite codimension necessarily prime?

I'm trying to answer the following question: Suppose that $R$ is an integral domain containing a field $K$. Then we may view $R$ as a $K$-vector space. Show that if R is finite dimensional as a K-...
3
votes
0answers
53 views

Factoring $x^n - 1$ and minimal polynomials

Let $gcd(n,q) = 1$ I'm trying to get to grips with factorising the polynomial $x^n - 1$ over $\mathbb{F}_q$. Firstly, it is a good idea to find an extension field containing all the roots of $x^n - 1$...