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.

learn more… | top users | synonyms (1)

0
votes
1answer
29 views

Proving equivalent formulations of Jacobson radical

I should start by saying I found this post Equivalent definitions of the Jacobson Radical which is about the same two formulations of the Jacobson radical but it didn't really to answer my question. ...
0
votes
1answer
89 views

Faithful representation of a $p$-group

Suppose $G$ is a nontrivial $p-group$. Let $H$ be the intersection of the center of $G$ and the set of elements in $G$ of exponent $p$. Let $\rho: G\rightarrow GL(V)$ be a representation. Show that if ...
1
vote
2answers
167 views
+100

Show that the sequence is exact

We have that $R$ is a commutative ring. Suppose that $0\rightarrow A\rightarrow B\overset{f}{\rightarrow} C\rightarrow 0$ and $0\rightarrow C\overset{g}\rightarrow D\rightarrow E\rightarrow 0$ are ...
0
votes
0answers
37 views

A module such that Ext$^i(M,A)=0$ for all $i$. [closed]

Let $A$ be a ring. Does it exist a module such that Ext$^i(M,A)=0$ for all $i$?
3
votes
1answer
55 views

$\chi : \text{Hom}(M,A) \otimes \text{Hom}(A,N) \to \text{Hom}(M,N) $

Let $M,N$ be $A$-modules. Consider the map. $$\chi : \text{Hom}(M,A) \otimes \text{Hom}(A,N) \to \text{Hom}(M,N) $$ $$f \otimes g \mapsto g \circ f $$ If $M,N$ are projective I do know that ...
0
votes
0answers
16 views

Hall $\pi$-subgroup property

Let $A$ be a normal subgroup of $G$ and $H$ be a Hall $\pi$-subgroup of $G$ Let $K$ be a Hall $\pi$-subgroup of $N_G(HA)$. I need to show that $K\leq HA$ and $KA = HA$ Since $[G : H] = [G : ...
2
votes
1answer
31 views

Subgroup of prime index

Let $G$ be a solvable group and $p$ is a prime such that $p\mid |G|$. Does there exist a subgroup $H$ of $G$ such that $[G:H]=p$ ?
2
votes
2answers
27 views

If $R_1$ is a division ring, any nontrivial homomorphism $\phi : R_1 \rightarrow R_2$ is injective

For rings $R_1, \: R_2$ let $\phi : R_1 \rightarrow R_2$ be a homomorphism. It's easy to show that $\ker \left (\phi \right)$ is an ideal for $R_1$ Now if we assume $R_1$ is a field (or a division ...
1
vote
2answers
34 views

Could we find an element on finite field? [closed]

Let $F$ be a finite field. Given an element $a^x$ in $F\setminus\{0\}$, could we find $a$?? I know that finding an integer $x$ is very hard problem (Discrete Logarithm Problem). However, I don't know ...
0
votes
0answers
23 views

Resultant of two single-variable polynomials via long division

I need to calculate the resultant of $Q=X^{10}+X^9 + \cdots + 1$ and $P= X^3+X^2+1$ by hand, and I already know it should be $23$. I'm obviously not gonna take the naive way via the coefficient ...
1
vote
2answers
45 views

Killing form is negative definite

Let $G$ be a compact connected semisimple Lie group and $\frak g$ its Lie algebra. It is known that the Killing form of $\frak g$ is negative definite. What about the Killing form $B$ of the complex ...
1
vote
0answers
53 views

Proof that $G/H_1$ is isomorphic to $G/H_2$ when $H_1$ and $H_2$ are normal groups of $G$

Let $H_1$ and $H_2$ be normal subgroups of a group $G$. I want to show, that $G/H_1$ is isomorphic to $G/H_2$ when $H_1$ and $H_2$ are normal groups. Why $H_1$ and $H_2$ should be conjugate subgroup ...
0
votes
0answers
28 views

Localization and completion under a strong hypothesis

This question is closely related to this one, but in my case I think the hypotheses are different. Let $(A,\mathfrak m)$ be a regular, local noetherian domain (the local ring at a smooth point of ...
5
votes
2answers
48 views

Finding $p(x)$ such that $\mathbb{Q}(\sqrt{1+\sqrt{5}})$ is ring isomorphic to $\mathbb{Q[x]}/\langle p(x)\rangle$

I am trying to find a polynomial $p(x)$ in $\mathbb{Q}[x]$ such that $\mathbb{Q}(\sqrt{1+\sqrt{5}})$ is ring isomorphic to $\mathbb{Q[x]}/\langle p(x)\rangle$. This is what I tried to do: Consider ...
-1
votes
0answers
14 views

Let G be a group with identity ε. If a, b ∈ Z and x ∈ G are such that x a = ε and x b = ε then show that x gcd(a,b) = ε. [duplicate]

I know the definition of the group is associative, inverse and identity. But, I have no idea where to start and how to solve it!
1
vote
3answers
48 views

Arithmetic Operations with Infinities in Real Analysis

Infinity is not a number , thus we cannot perform the usual arithmetic operations that we do with real numbers This is the usual reason given when asked why we can't perform the usual arithmetic ...
0
votes
0answers
26 views

Finite group $G$ satisfies that for all positive integer $m$ number of solutions of $x^m=e$ are at most $m$ is cyclic. [duplicate]

Finite group $G$ satisfies that for all positive integer $m$ number of solutions of $x^m=e$ are at most $m$ is cyclic. It is exercise in A First Course in Abstract Algebra by Fraleigh. The book ...
0
votes
0answers
13 views

Injective map from the Cartesian product of two sylow-p-subgroups into the group.

Let $G$ be a group of order 148. Show that $G$ is not simple. The given solution goes as follows: $148 = 4 × 37$. By Sylow’s theorem, it has at least one subgroup $P$ of order 37. If $P'$ is ...
1
vote
0answers
49 views

Genetic algebras; what should a mathematician know about them?

I recently learned that genetic algebras are a thing, which means there is a link between abstract algebra and genetics. Question: What should I (wearing my mathematician's hat) know about them? ...
9
votes
1answer
94 views

$\mathbb{Q}(\sqrt{23})$ is not a Euclidean number field.

The problem I'm facing is that of the tittle: Problem. Prove that $\mathbb{Q}(\sqrt{23})$ is not a Euclidean number field. Since $23\not\equiv 1\pmod{4}$, it must be shown that ...
4
votes
1answer
25 views

Finding a homomorphism from a subset of the fractions to the ring $\mathbb{Z}_p$.

I'm practicing using the First Isomorphism Theorem for rings. Here is a question I got stuck on. Let $p$ be prime and let $T$ be the set of rational numbers (in lowest terms) whose denominators ...
1
vote
1answer
35 views

Quotient ring $R/I$, $R= \mathbb{Z}[\sqrt{-10}]$, $I =(\sqrt{-10})$

To show $I =(\sqrt{-10})$ is not prime in $R=\mathbb{Z}[\sqrt{-10}]$, there is a direct method, i.e. show that $2$ and $5$ are not in $I$ but their product $10$ is. However, I struggled to prove the ...
0
votes
1answer
27 views

Let $A$ be a ring, $M''$ and $P$ be $A$-modules and $f:P\rightarrow M''$ a homomorphism…

I need some help with this problem, it seems easy, but I don't get it. Let $A$ be a ring, $M''$ and $P$ be $A$-modules and $f:P\rightarrow M''$ an homomorphism. Suppose that for each $A$-modules $M$ ...
0
votes
2answers
53 views

Why is $\mathbb{Q}(\sqrt{2}\sqrt[3]{3}) \subset \mathbb{Q}( \sqrt{2},\sqrt[3]{3})$

Why is $\mathbb{Q}(\sqrt{2}\sqrt[2]{3}) \subset \mathbb{Q}( \sqrt{2},\sqrt[3]{3})$ "obvious"? My book states this as obvious, but then proves the opposite inclusion. I would have thought that ...
2
votes
1answer
74 views

Transitivity-like Results in Group, Ring, Module, Field and Galois Theory [closed]

I am reading Michael Atiyah and Ian Macdonald's Introduction to Commutative Algebra. On page 28, Proposition 2.16 says: Suppose $A,B$ are rings, $N$ is a finitely generated $B$-module, $B$ is ...
1
vote
1answer
20 views

Conjugacy of Hall $\pi$-subgroups of a group $G$

A group is called a $C_\pi$-group if there exists a Hall $\pi$-subgroup and any two Hall $\pi$-subgroups are conjugate. Let $G$ be a group such that $N$ is a normal $C_\pi$-subgroup of $G$ and ...
-1
votes
0answers
20 views

$Aut(S_n)=Inn(S_n)$ for all natural number n except 6. [duplicate]

Since $G/Z(G)$ is isomorphic to $Inn(G)$ and $Z(Sn)=1$, If $Aut(S_n)=Inn(S_n)$, then $S_n=Aut(S_n)$. How can I show that $Aut(S_n)=Inn(S_n)$ for except $n=6$?
5
votes
1answer
74 views

Given two algebraic conjugates $\alpha,\beta$ and their minimal polynomial, find a polynomial that vanishes at $\alpha\beta$ in a efficient way

Inspired by this question, I was wondering about the following problem. $\alpha,\beta,\gamma,\ldots$ are the roots of an irreducible polynomial over $\mathbb{Q}$. How to compute the coefficients ...
-2
votes
0answers
40 views

Does charR=0 imply that R is a field? [closed]

What are the implications of charR=0? If D is an ID then charD=0 or p. I don't know if the converse is true.
0
votes
1answer
39 views

Number of elements in $U(n)$ with order dividing $n-1$ [closed]

Please suggest a solution to this problem: Let $p^2\ |\ n$ for some prime $p$. Show that there may exists at most half of elements $a$ in the multiplicative group of integers modulo $n$, such that ...
2
votes
1answer
26 views

Q: $\alpha: H \rightarrow \operatorname{Aut}(H)$ a nontrivial homomorphism. Must $H \rtimes_{\alpha} H \ncong H \times H$?

$H$ is a group and $\alpha: H \rightarrow \operatorname{Aut}(H)$ is a nontrivial homomorphism. Does $H \rtimes_{\alpha} H \ncong H \times H$ necessarily? This is a follow-up to this thread. ...
1
vote
0answers
13 views

The Variety X(K) as a subset of the analytification of X

I am working with Sam Payne's artical Analytification is the limit of all tropicalizations, and because of my limited understanding of analytification it gives me some difficulties. The article: ...
0
votes
0answers
27 views

Explicit matrix representation of an algebraic extension

This may be considered an extension of this question. Let $\mathbb{F}$ be a field, and let $p(X)\in\mathbb{F}[X]$ be an irreducible polynomial. Let $\mathbb{F}_p$ be the extension of $\mathbb{F}$ by ...
-1
votes
0answers
35 views

Showing that $1-a$ has a multiplicative inverse in a ring $R$ [duplicate]

Question Let a belong to a ring R with unity and suppose that $a^{n}=0$ for some positive integer n. Prove that $1-a$ has a multiplicative inverse in $R$. Hint: $$\left ( 1-a \right ...
0
votes
1answer
48 views

Extension of DVRs and uniformizers

Let $(A,\mathfrak m)$ be a regular, Noetherian, local, domain of dimension $2$ and consider a prime ideal $\mathfrak p\subset A$ of height $1$. Moreover let $\hat{A}$ be the completion of $A$ with ...
0
votes
2answers
39 views

Power of a polynomial in Galois field

Let $f(x) \in GF[q](X)$, where $q = p^m$ and $p$ prime. Is the following true? $$f^{p^m}(X) = f(X^{p^m}).$$ I tried to prove the assertion above and got stuck at the following: $$ \begin{align} ...
0
votes
1answer
41 views

Maximal ideals in a ring of sets

A ring $R$ is called Boolean if $x^2 = x$ for all $x \in R$. It follows that Boolean rings have characteristic $2$ and are commutative. Let $S$ be a non-empty set, then $P(S)$ with $A + B = (A - B) ...
1
vote
1answer
37 views

$n$th root of power series when its coefficients are from a field with positive characteristic

Let $k$ be algebraically closed field of characteristic $p>0$. Let's consider a power series $f(x,y)\in k[[x,y]]$. Under what conditions (on $n$, $f$, ...) there exists $g(x,y)\in k[[x,y]]$ such ...
2
votes
1answer
22 views

How to find normal subgroups from a character table?

I know that normal subgroups are the union of some conjugacy classes Conjugacy classes are represented by the the columns in a matrix How could we use character values in the table to determine ...
1
vote
0answers
49 views

How many normal groups does $\mathbb{Z}_n \times \mathbb{Z}_m$ have?

I am currently studying abstract algebra, and I found this problem. I can determine the number of elements and the maximum order of the same. I need to find the subgroup H to apply the theory, but how ...
2
votes
1answer
38 views

$p$-groups have normal subgroups of each order [duplicate]

Suppose $|G|=p^n$. Then $G$ has a normal subgroup of order $p^m$ for every $0\le m\le n$. By induction. It is clearly true for $n=0$. Now suppose $k<n$ and $H_i$ is a normal subgroup of $G$ of ...
1
vote
3answers
37 views

Trivial intersection of quotient subgroups

Suppose that $H/P$ and $M/P$ are two subgroups of a group $G/P$ such that intersect trivially i.e. $(H/P) \cap (M/P) = \{1_{G/P}\}$ = $\{P\}$. Is it true that $H \cap M = P?$
1
vote
0answers
17 views

Valuation fields and linear disjointness

I've got a question about linear disjointness of extension fields. Here I refer to the definition that Wikipedia gives: Two algebras $A$, $B$ over a field $k$ inside some field extension $\Omega$ ...
0
votes
0answers
20 views

Algebraic K-theory: induced maps

Let $f:A \to B$ be a homomorphism of rings. One way to define algebraic $K$ theory ($K_0$ group) is to consider the Grothendieck group of isomorphism classes of all finitely generated projective ...
1
vote
1answer
43 views

Representation of elements in a field

Let $A$ and $B$ be integrals domains, such that $A$ is integral over $B$. Writing $K(A)$ for the field of fractions, suppose that $K(A)$ is generated over $K(B)$ by a single element in $A$, say ...
0
votes
1answer
25 views

Dimension of a direct sum of characters (example with $S_3$)

Here is the character table of $S_3$: I was wondering how one can determine the dimension of for example the sign character $sgn$. Could we get it from the character table? Also, if we define $A$ ...
-1
votes
1answer
58 views

Example of an ideal which is not principal in the ring $\mathbb{Z} [x]$ [duplicate]

Give an example of an ideal in the ring $\mathbb{Z} [x]$ is not principal. What kind of example would be the easiest?
0
votes
1answer
23 views

Why if $Z/m\oplus Z/n = Z/mn$ then $(n,m)=1$ [duplicate]

Prove that if $Z/m\oplus Z/n = Z/mn$ then $(n,m)=1$. I have proved the converse, but here there is something I am missing. Hints instead of full answers are appreciated. Thanks.
0
votes
0answers
23 views

If $H$ and $K$ are nilpotent normal subgroups then $C(HK)$ is non trivial

I know that this follows from the fact that $HK$ is nilpotent but maybe there is an easier way to proof this? I wanted to show that there is an Element in $HK$ that commutes with $H$ and $K$. I ...
4
votes
2answers
81 views

The rigid additive tensor category freely generated by an object

I've found myself to be absolutely mystified by something in Deligne and Milne's notes on Tannakian categories. Namely, on p. 16 they are showing that there is a rigid additive tensor category ...