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)

5
votes
1answer
33 views

$F[[T]] \times F[[1/T]]$, fundamental domain.

Let $p$ be a prime number. Here is a link which shows how to see that $$(\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T)))/\mathbb{F}_p[T, 1/T]$$is compact using an adelic result. (Here $\mathbb{F}_p[T, ...
2
votes
1answer
51 views

Example of non-commutative ring with exactly 2014 two sided-proper ideals.

Find a non-commutative ring with exactly 2014 two sided-proper ideals. Find a ring with exactly 2014 pairwise non-isomorphic irreducible modules. If it was the commutative ring i would have ...
0
votes
2answers
20 views

Definition of algebraically closed field.

A field F is algebraically closed if every non constant polynomial in F[x] has a root in F. Is this the right definition? I am wondering if only one root in F and the rest of the roots not in F can ...
0
votes
1answer
380 views

degree of the extension galois

I have a problem with the solution of the tasks of abstract algebra.Help please. $F=({\bf Q};+;\cdot),K=({\bf R},+;\cdot)$. Determine the degree of the extension $F_K^* (\sqrt 2,\sqrt 3):F]$. ...
-1
votes
2answers
33 views

From group isomorphisms to algebra isomorphisms

Let $A$ be an algebra and let $A^{\ast}$ be the subset of units (that is, invertible elements) of $A$. Then $A^{\ast}$ is a group under the multiplication of $A$. Let $f^{\ast}:A^{\ast}\to A^{\ast}$ ...
-1
votes
0answers
28 views

How to compute $Z_n \times Z^*_m$

How to compute $Z_n \times Z^*_m$.(In journal Multiplicative Properties of Set Residues) say by chinese remainder theorem, may be thought of as $\left\{a \in Z_{mn}:(a,m)=1\right\}$. where $Z^*m$ unit ...
1
vote
2answers
21 views

Generators of the group of invertible elements of the ring $\mathbb{Z}_{14}$—are they multiplicative or additive?

When I was asked to find the generators of the group of invertible elements of the ring $\mathbb{Z}_{14}$, which are denoted as $\phi(14)$, I did not realize whether the generators are multiplicative ...
2
votes
1answer
35 views

Why is the commutator group a subgroup?

I am in Intro to Algebra, and have a question regarding the commutator subgroup. I am a bit confused with the premise, though, with how the set is a subgroup in the first place. Let $C$ be the set of ...
0
votes
1answer
15 views

Number of equivalence Relations containing $(1,2)$

Find the number of equivalence Relations on the Set $A=\{1,2,3 \}$ which contains the Element $(1,2)$. My Try: Since $(1,2)$ is to be included, so is $(2,1)$ since the Relation should be Symmetric ...
1
vote
3answers
51 views

Looking for help to understand example of Group

I am looking for someone to help me to understand what is going on in the following example, from Hersteins "Topics in Algebra". It says, Let $G$ be the set of all $2*2$ matrices $$\begin{pmatrix} ...
2
votes
2answers
90 views

When does the regularity of $A$ implies the regularity of $A[w]$?

Let $A$ be a commutative noetherian ring (I do not mind to assume that $A$ is a UFD), and assume that $A$ is regular. Recall that a commutative noetherian ring is called regular if all its ...
0
votes
1answer
21 views

Unique factorization domain: $\mathbb{Z}_{n}[x]$

How to determine all $n\in\mathbb{N}$ such that $\mathbb{Z}_{n}[x]$ is a unique factorization domain? I am guessing that this would be true for all primes, since $\mathbb{Z}_n$ is a UFD when $n$ is ...
7
votes
0answers
98 views

Subring of $\mathcal O(\mathbb C)$

Let $\mathfrak A \subset \mathcal O(\mathbb C)$ be the subring generated by the nowhere zero analytic functions $f: \mathbb C \to \mathbb C$. Does we have a precise description of $\mathfrak A$ ? Is ...
-2
votes
1answer
55 views

Ideal in $\mathcal O(\mathbb C)$

Let $\mathfrak {I}$ the ideal generated by all the holomorphic functions which are never zero. Question : is $\mathfrak {I} = \mathcal O(\mathbb C)$ ?
21
votes
9answers
1k views

What's a group whose group of automorphisms is non-abelian?

I recently attended an interview for admission to graduate programs in Mathematics. The interviewing professor asked me a question - Tell me a group whose group of automorphisms is non-abelian. ...
6
votes
0answers
40 views

$y^2 = x^3 - 26$, exist ideal satisfying conditions?

For the solution $(x, y) = (3, 1)$ of $y^2 = x^3 - 26$, does there necessarily exist an ideal $I$ of the integer ring $\mathbb{Z}[\sqrt{-26}]$ of $\mathbb{Q}(\sqrt{-26})$ such that $(y + \sqrt{-26}) = ...
16
votes
0answers
221 views
+100

Meromorphic functions on $U^2 = T^3 + 1$, genus.

Let $k$ be a field of characteristic $\neq 2$, and consider the quadratic extension $F$ of $k(T)$ generated by $\sqrt{T^3 + 1}$. What is/how do I find the genus of $F$? The progress I have so far: ...
3
votes
1answer
27 views

If $G$ is a group of order $2^nm$, where $m$ is odd and $(m-1)!<2^n$, show that $G$ is not simple.

If $G$ is a group of order $2^nm$, where $m$ is odd and $(m-1)!<2^n$, show that $G$ is not simple. I started out by trying to prove this using the Sylow theorem, but it led nowhere. I was able to ...
2
votes
2answers
44 views

Finitely generated abelian group with certain properties

Problem Characterize all finitely generated abelian $G$ such that every proper subgroup of $G$ is cyclic, $G$ contains exactly two proper subgroups, and for each pair of subgroups $S$,$T$ in $G$ ...
0
votes
1answer
35 views

Acting algebraically

Let $G$ be a group that acts on some non empty set $V$. What does it mean that $G$ acts algebraically on $V$? I am well aware of the definition of being algebraic. But I cant find the definition ...
0
votes
0answers
12 views

every $F$-algebraic homomorphism of $K$ is $1-1$ and onto? [duplicate]

Suppose that $K|F$ is a field extension of finite degree. We know that every $F$-algebraic homomorphism of $K$ is $1-1$ and onto. Also we know that every finite field extension is algebraic extension. ...
2
votes
1answer
27 views

is $\mathbb{Q} ^{|\mathbb{R}|} ‎\times‎ \{1 \}$ divisible subgroup of $ \mathbb{Q} ^{|\mathbb{R}|} ‎\times‎ \mathbb{Z}_2$?

According to Unit Groups of Classical Rings by Karpilovsky, p.107 we know that: If $F$ is a real-closed field, then $F^*‎\simeq‎ \mathbb{Q} ^{|F|} ‎\times‎ \mathbb{Z}_2$. Now, we know that ...
1
vote
0answers
25 views

Is it true that $M\otimes_A F\simeq M^{(I)}$?

Let $A$ be an $R$-algebra ($R$ is a commutative ring with identity $1_R$) and suppose $F$ is a left free module over $A$. Is it true that $$M\otimes_A F\simeq M^{(I)}$$ for any right module $M$ over ...
2
votes
1answer
30 views

Another approach to $A/I\otimes_A A/J\simeq A/(I+J)$?

The same question appears here $A/ I \otimes_A A/J \cong A/(I+J)$ however I'm looking for a different approach. Let $A$ be an algebra, $I$ a right ideal and $J$ a left ideal. I'd like to show ...
0
votes
2answers
32 views

Which of the following about a permutation is correct?? (CSIR-2015, June)

Let $\sigma:\{1,2,3,4,5\}\rightarrow\{1,2,3,4,5\}$ be a permutation (one-to-one and onto function) such that $$ \sigma^{-1}(j)\le \sigma(j) \quad\text{for all $j$ such that }1\le j\le 5. $$ Then ...
4
votes
1answer
231 views

What is the intuition of conjugacy classes?

How can I fully understand what are conjugacy classes are in groups? I know the definition, that $a$ and $b$ are conjugate if $gag^{-1}=b$ for some $g\in G$. But what is the intuition? Using a ...
0
votes
0answers
14 views

Normal Submagma?

Is there a definition of normal submagma? visit https://en.wikipedia.org/wiki/Magma_(algebra) For normal sub-quasi-group I found two: A sub-quasi-group $H$ is called normal if there exists a normal ...
2
votes
1answer
37 views

Subgroup proof verification.

Let $G$ be an abelian group, K is a fixed positive integer. $H$={$a\in$ $G$ $|$ $|a|$ divides K} . Prove that $H$ is a subgroup of $G$. My way of proving (Let me know how I could make it better or ...
6
votes
0answers
152 views
+50

Unsolvability of a Quintic and its link with “Simplicity” of $A_{5}$

At the outset I must mention that I don't have a fairly working knowledge of Galois Theory (but do have some idea of group theory in the sense that I can understand normal subgroups). I read the ...
4
votes
2answers
46 views

Groups of the from $gMg$ in a monoid where $g$ is an idempotent

Let $(M, \cdot)$ be a finite monoid with identity $e$. It is easy to see that $gMg = \{ gxg : x \in M \}$ forms a monoid with identity $geg = g$ if $g$ is an idempotent. If $gMg$ contains no ...
2
votes
0answers
35 views

Application of the structure theorem for finitely generated modules over a PID

I've been trying to figure out the correct way to do this and I'm not sure I've got it quite right. $\mathbb{Z}^4/S$ where $S = \{(4, 12,20,8),(6,30,18,12),(4,16,16,8),(8,40,24,16) \}$ So, $M_0 = ...
3
votes
1answer
559 views

Finding a basis for a submodule

Let $F$ be the $\mathbb{Z}$ -module $\mathbb{Z}^{3}$ and let $N$ be the submodule generated by {$(4,-4,4),(-4,4,8),(16,20,40)$}. Find a basis $\left\{ f_{1},f_{2},f_{3}\right\}$ for $F\textrm{ ...
2
votes
1answer
67 views

Finding a binary operation on $\{1, \dots, n\}$ so that each $k$ has exactly $k - 1$ left inverses

What is an example of a binary operation on the set $\{1, \dots, n\}$ so that each element $k \in \{1, \dots, n\}$ has respectively $k-1$ left inverses? I have been trying various combinations ...
1
vote
1answer
14 views

Explicitly compute the trace for the tautological representation of $D_4$ of $\mathbb{R}^2$.

Fix a finite dimensional representation $\rho: G \longrightarrow GL(V)$ of $G$. Its trace is defined as the function $tr:G \longrightarrow F$ defined by $tr(g) = tr(\rho(g))$. Explicitly compute ...
6
votes
1answer
43 views

Quadratic field, $O_K/\mathfrak{p} = \mathbb{F}_p$, $O_K/pO_K$ is a finite field of order $p^2$.

Let $K$ be a quadratic field $\mathbb{Q}(\sqrt{m})$ where $m$ is a square free integer, and let $p$ be a prime number which does not divide $2m$. Where can I find a reference to a proof of the ...
0
votes
2answers
76 views

Hardcore Abstract Algebra Book Request

I am going to start learning Abstract Algebra soon. I was originally going to start with Dummit and Foote, but I am starting to abandon that idea. I want to use a "hardcore" algebra book. I don't mind ...
0
votes
1answer
29 views

Non-zero maps between modules

Let $R$ be a ring and $M,N$ two $R$-modules such that there exists a non-zero map $\psi : M \to N$. Is it true that there exists a non-zero map $\varphi: N \to M$ ? I do not have any particular ...
4
votes
0answers
36 views

The ordinals as a free monoid-like entity

Let $\kappa$ denote a fixed but arbitrary inaccessible cardinal. Call a set $\kappa$-small iff its cardinality is strictly less than $\kappa$. By a $\kappa$-suplattice, I mean a partially ordered set ...
5
votes
2answers
69 views

$L$-function, easiest way to see the following sum?

What is the easiest way to see that$$\sum_{(m, n) \in \mathbb{Z}^2 \setminus \{0, 0\}} (m^2 + n^2)^{-s} = 4\zeta(s)L(s, \chi)?$$Here $\chi$ is the homomorphism $(\mathbb{Z}/4\mathbb{Z})^\times \to ...
0
votes
1answer
79 views

Group as a $\mathbb Q$-vector space

Let $G$ be a torsion free abelian group of having $n$ number of maximally rationally independent elements $r_{1}, r_{2}, ..., r_{n}$ and assume that $G$ is not finitely generated. Is this correct ...
0
votes
1answer
43 views

Follow up question to finding primes $p$ such that $f(x)=x^6 - x^3 +1$ factors (in various ways) in $\mathbb{F}_p$

I asked this question yesterday, however, I am not sure how to compare the solution given on this site to the "worked example" solution as in my notes. $\textbf{Problem Statement:} $ Let $f(x)= x^6 - ...
7
votes
1answer
86 views

What is the Galois group of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$, where $\zeta$ is a $p^r$th root of unity?

Let $\zeta$ be a $p^r$-th root of unity and $\mathbb{Q}_p$ the p-adic numbers. I would like to know the Galois automorphisms of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$. I already know, the degree of ...
2
votes
1answer
58 views

Example of subgroup of $\mathbb Q$ which is not finitely generated

I was looking for a proper subgroup of $\mathbb Q$ which is not finitely generated under the addition operation. We know every finitely generated subgroup of $\mathbb Q$ is cyclic. For a proper ...
1
vote
0answers
24 views

Galois group isomorphic to $\mathfrak{S}_5$.

Let $f(x) \in \mathbb{Q}[x]$ an irreducible polynomial of degree $5$ and with splitting field $K \supset \mathbb{Q}$. If $\mathbb{Q}(\sqrt{7})$ and $\mathbb{Q}(\sqrt{11})$ are subfields of $K$,is ...
0
votes
1answer
33 views

Ring Structure for Non Commutative Groups: Is there a grander reason for Abelian requirements?

So I was considering the following idea. Let a generalized Ring $gR$ be a set equipped with two operations $$u_1, u_2$$ Such that $gR$ is a with the operation $u_1$ and the operation $u_2$ ...
4
votes
4answers
75 views

what is the greatest integer that divides $p^4-1$ for every prime number p greater than 5

what is the greatest integer that divides $p^4-1$ for every prime number p greater than 5(this is a gre subject math problem) I think that $p^4-1=(p^2+1)(p-1)(p+1)$,so 8 must divide all the $p^4-1$ ...
1
vote
0answers
37 views

Group isomorphism exercise

I am trying to solve the following problem: Let $G$ be a finite group such that there exists an isomorphism $\phi:f:G/Z(G) \to \mathbb Z_3 \oplus \mathbb Z_3$ and $x \in G$ such that ...
3
votes
1answer
55 views

Exercise from Serre's “Trees” - prove that a given group is trivial

In Serre's book "Trees" on page 10 the following exercise is given: Show that the group defined by the presentation $$x_2x_1x_2^{-1}=x_1^2, \hspace{7pt} x_3x_2x_3^{-1}=x_2^2, \hspace{7pt} ...
0
votes
0answers
26 views

Are there infinite-dimensional, artinian C*-algebras?

A ring is artinian if it has no infinite descending chains of ideals. Of course finite-dimensional algebras are artinian. I'm wondering if it's possible to have an artinian C*-algebra (or Banach ...
0
votes
1answer
54 views

Prove that $\mathbb{Q}{[\sqrt3]} = \lbrace a + b\sqrt3: a, b \in \mathbb{Q} \rbrace$ is a subfield of $\mathbb{R}$

Prove that $\mathbb{Q}{[\sqrt3]} = \lbrace a + b\sqrt3: a, b \in \mathbb{Q} \rbrace$ is a subfield of $\mathbb{R}$ My answer: Must show: i) $0 \in \mathbb{Q}{[\sqrt3]}$ ii) $1 \in ...