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)

2
votes
3answers
21 views

Show that the additive inverse condition can be replaced by $0v = v$ for all $v \in V$

In the definition of a vector space, the additive inverse condition requires that for every $v \in V$ (where $V$ is a vector space over $\mathbf{F} = \mathbb{R}$ or $\mathbb{C}$), there exists ...
3
votes
0answers
31 views

Which elements of $su(n)$ commute with those of a subalgebra $su(2)$

Given a subalgebra $su(2) \subset su(n)$ , how many generators of $su(n)$ commute with any element in the subalgebra $su(2)$? I know that there are at least $n-2$ elements in $su(n)$ satisfying this ...
1
vote
1answer
31 views

Resources on exterior algebra, wedge product, geometric product and tensors

I would like to learn exterior algebra, wedge product and geometric product along with their applications in physics. Is there a good source you can recommend? Should I study differential geometry in ...
1
vote
0answers
58 views

Prove that if $3^\frac{F_n-1}{2} \equiv -1 \pmod {F_n}$, then $F_n$ is prime

$F_n = 2^{2^n} + 1$ is a Fermat Number. Here is my attempt. We square each side of the congruences we get $$3^{F_n-1} \equiv 1 \pmod {F_n}$$ Now I already know that whenever $m \neq n$ then $ ...
3
votes
1answer
35 views

$\mathbb{Z}/p^n$ is an injective module over itself. [duplicate]

For any prime power $p^n$, prove that $\mathbb{Z}/p^n$ is an injective module over itself.
0
votes
0answers
26 views

A group which satisfies these conditions

I am looking for a group which satisfies the following conditions- $1.$ $G \le U_1(\mathbb{Z}G)$ , where $U_1(\mathbb{Z}G)$ is the set of normalized units of $\mathbb{Z}G$ i.e. $U_1(\mathbb{Z}G)$ ...
1
vote
0answers
34 views

Group Ring over commutative ring

$R$ is commutative semiprime ring, $(R,+)$ abelian group without torsion. Then $RG$ is semiprime. Proof by contradiction. If $x$ is nilpotent element in $RG$, then $x=r_1g_1+r_2g_2+...r_ng_n$. ...
1
vote
1answer
20 views

Universal property question

I am not sure how I can draw a commutative diagram, so I will do my best to describe it verbally. So, suppose $f_1,f_2:G\to K$ be (group, field, ring)homomorphisms. I want to claim the following: ...
1
vote
1answer
15 views

Representation matrix for modules map

Here $S=\mathbb Q[x,y]$, and we define $\oplus Se_i$ to be a $S$-free module with basis $\{e_1,e_2,e_3\}$. Define a map from $\oplus Se_i$ to $S$ by $e_1\to x^2$, $e_2\to xy+y^2$, $e_3\to y^3$. Is the ...
2
votes
1answer
63 views

Why are structures with no relations called algebras?

"If [a given structure] A has no relations it is termed an algebraic structure, or simply an algebra" - Wolfgang Rautenberg, A Concise Introduction to Mathematical Logic, 3rd edition, page 42. I ...
1
vote
0answers
23 views

How is $K(x)$ pronounced, where $K$ is a field?

Let $K$ be a field. The ring of polynomials $K[x]$ is pronounced "$K$ adjoin $x$", right? How is the field of rational functions $K(x)$ pronounced? (Sorry if this is a silly question. I am ...
0
votes
1answer
33 views

Basis of a Z-module

I think I might know how to start this problem but I'm not sure how to finish. Here is the statement: Determine a basis for the ℤ-module of integer solutions to the following system of equations: ...
0
votes
0answers
35 views

Fundamental Domain of ${\mathbb Z}^2$ to ${\mathbb R}^2$

Find a fundamental domain for the action of $\mathbb Z^2$ on $\mathbb R^2$ by translation A fundamental domain is the nodes $(0,0),(1,0),(0,1)$ and the edges which connect them Is there a better way ...
1
vote
1answer
48 views

Proving the number of solutions in a finite field

Let F be a finite field containing $q$ elements. Let $a \in F$ be nonzero. If n divides $q -1$ prove that $x^n - a = 0$ has either n solutions in F or no solutions in F. I am unsure of how to begin ...
0
votes
1answer
27 views

Can't understand a proof of maximal ideal

The question is Let $R = \{a+b\sqrt{2} \mid a,b \text{ integers}\}$. Let $M = \{a+b\sqrt{2} ~|~ 5|a \text{ and } 5|b\}$. Prove $M$ is a maximal ideal. We already had the knowledge $M$ is an ...
0
votes
2answers
22 views

Equation of Class of a group of cardinality 60. [on hold]

Let be $G$ a group of cardinality 60. Suppose, furthermore that equation of class is $60=1+12+12+15+20$. Show that the unique normal subgroup of $G$ are {id} and $G$. Any help, plis..
1
vote
1answer
31 views

prime ideal of $\mathcal O_K$ splits completely in tower of Galois extensions

The following exercise is taken from D.A.Cox's book "Primes of the form $x^2 +ny^2...$" He uses this in order to prove some intermediate steps before the proof of the main theorem in chapter 5, which ...
0
votes
1answer
14 views

σ -algebra by choosing sets, where either the set or its complement is countable: is the complement countable?

I am reading Schilling's “Measures, integrals and martingales”, where on page 15 he constructs a $σ$-algebra, according to: $$ \mathcal{A} = \lbrace A \subset X: \# A \leq \# \mathbb{N} \quad ...
0
votes
2answers
40 views

What's the difference between finite and finitely generated algebras

I didn't understand the difference between the two definitions: I thought the definition of $B[a_1,\ldots,a_n]$ is exactly the one in the item (b), i.e., $B[a_1,\ldots,a_n]=Ba_1+\ldots+Ba_n$. I ...
3
votes
1answer
33 views

Set of units in ring a group?

I am supposed to prove that given a commutative ring $R$, the set of units $R^{\times}$ is a group. I checked the axioms of a group and it all came down to noting that if $a,b\in R^{\times}$, then ...
0
votes
1answer
66 views

Nilpotent elements in group algebra

Suppose $FG$ -- is group algebra and $F$ is field with characteristic $p>0$. $G$ - is finite $p$-group. Thus, it's clear that $(e-g)$ is nilpotent. But how to show that $(e-g)g_1$ is nilpotent for ...
1
vote
0answers
92 views

How to prove that an ideal can not be generated by 2 elements

In Kunz's "Introduction to commutative algebra and algebraic geometry", page 137-139, particular monomial affine curves are described. Here is the link. In case the curve is not an ideal ...
-4
votes
0answers
19 views

normal extension field [closed]

How do you know if these are normal extensions of the rational numbers? a. Q(√3) b. Q(∛3)
1
vote
0answers
37 views

Action of matrix on symmetric products

Suppose that $M : V \to V$ is a linear map of a finite-dimensional vector space. This induces a linear map $M_n : \operatorname{Sym}^n(V) \to \operatorname{Sym}^n(V)$ for any $n \geq 1$. Is there a ...
-3
votes
0answers
25 views

Galois Theory, cyclic [closed]

A finite normal extension K of a field F is cyclic over F is G(K/F) is a cyclic group. Show that if K is cyclic over F and E is a normal extension of F, where F≤E≤K, then E is cyclic over F and K is ...
1
vote
1answer
22 views

Quotient by power of maximal ideal

Suppose $R$ is a commutative ring (but see the edit portion below) and $\mathfrak{m}$ is a maximal ideal of $R$ such that $|R/\mathfrak{m}|<\infty$. Also assume that $k$ a positive integer. Is ...
0
votes
0answers
39 views
0
votes
0answers
9 views

$U_1(\mathbb{Z}G)$ is a finitely generated FC-group.

If each member in support of an element in $\mathbb{Z}G$ is centralized by a subgroup of finite index in $G$, then why does it imply that $U_1(\mathbb{Z}G)$ is a finitely generated FC-group., where ...
0
votes
1answer
32 views

how to show that an ideal is convex [closed]

I need to show that the ideal $J=(i)$ in $C(\mathbb R)$ where $i$ is the identity function, $C(\mathbb R)$ is the ring of all continuous functions on the real numbers, is a convex ideal.
0
votes
1answer
22 views

definition of Artin map

Let $L/K$ be an unramified Abelian extension. Then the Artin symbol $ \left ( (L/K) / \mathfrak p \right) $ is defined for all prime ideals $\mathfrak p$ of $\mathcal O_K$. (because in an Abelian ...
0
votes
0answers
20 views

When does each action corresponds to a homomorphism and an anti-homomorphism.

If I adopt for function evaluation and function composition the convention $f(x)$ and $(f\circ g)(x) = f(g(x))$, and if $G$ is a left group action on some set $X$, then to this left action their ...
-1
votes
2answers
98 views

Should an object in the category always be a formal mathematical structure?

Today I heard during a popular lecture about the applications of category theory, than an object in the category should always be some kind of mathematical structure e.g. a set, ring, monoid, etc. So ...
-1
votes
0answers
33 views

If $A$ is a submodule of $B$ and $B$ is a submodule of $C$, is $A$ a submodule of $C$?

Let $C$ be a commutative ring (with 1, if this matters). If $A$ is a submodule of $B$ and $B$ is a submodule of $C$, is $A$ a submodule of $C$? I can't really prove that it is true because it is ...
1
vote
1answer
41 views

Problems with understanding the proof of noetherian ring

If $M$ is an $R$-module, the the following are equivalent: 1. M is finitely generated 2. M satisfies the ascending chain condition 3. Every non-empty set of submodules of M contains at least one ...
3
votes
1answer
47 views

$[U \cap H : G \cap H] \le [U :G]$

Is this always true that $[U \cap H : G \cap H] \le [U :G]$ , where U is a group and $G,H$ are subgroups of $U$? My trials for $\mathbb{Z}$ were giving affirmative answer but how to prove it, if it ...
-2
votes
0answers
32 views

a question about abstract algebra,prove that the ring is commutative. [duplicate]

(1)A ring R is a booleean ring if for every $a\in R$,$a^2=a$. Show that every Boolean ring is a commutative ring. (2)Let R be a ring,where $a^3=a$ for all $a\in R$.Prove that R must be a commutative ...
0
votes
0answers
31 views

Let $R$ be an Euclidean ring. Prove that $\operatorname{lcm}(a,b) = \dfrac{ab}{\gcd(a,b)}$ for $a,b \in R.$

Let $R$ be an Euclidean ring. Prove that $\operatorname{lcm}(a,b) = \dfrac{ab}{\gcd(a,b)}$ for $a,b \in R.$ Does this proof work? Any criticism appreciated: We rewrite $\operatorname{lcm}(a,b) = ...
7
votes
0answers
41 views

The difference between the ring version and module version of Chinese Remainder Thereom.

Chinese Remainder Theorem for Commutative Rings If $R$ is a commutative ring with $1$ and $I, J$ are ideals of $R$ that are pairwise coprime or comaximal (meaning $I + J = R$), then $IJ = I \cap J$, ...
0
votes
0answers
11 views

Criterions for $U_1(\mathbb{Z}G)=G$ i.e. units to be trivial in $\mathbb{Z}G$

Notation- $U_1(\mathbb{Z}G)$ here means units of $\mathbb{Z}G$ with augmentation $1$, i.e. $u=\sum_{g\in G} a(g)g \in U(\mathbb{Z}G)$ such that $\sum_{g \in G} a(g)=1$ 1) I have done theorem by ...
1
vote
1answer
42 views

Why $R/AB$ is cyclic?

Why $R/AB$ is cyclic when $A,B$ are ideals of $R$? I know a cyclic module is a module that can be written as $Rm$ and $Rm$ is isomorphic to $R/Ann(m)$. However, I can't see why the module is ...
3
votes
1answer
28 views

Factor into primes in Dedekind domains that are not UFD's?

Does it make sense to factor numbers into prime numbers in Dedekind domains that are not unique factorization domains? I can't really see how it would make sense.
0
votes
1answer
23 views

Element order proof [closed]

$n\in \mathbb{Z}$ and $\overline{a}\in U(\mathbb{Z}_n)$ order is $kl$. Prove that $\overline{a}^k$ order is $l$. Any ideas on how to approach this? It seems to follow straight from power definition. ...
2
votes
2answers
38 views

Prove that $\alpha$ and $\beta$ are homomorphisms from $F(\mathbb{R})$ onto $\mathbb{R}$

Let $\alpha : F(\mathbb{R}) \to \mathbb{R}$ be defined by $\alpha(f)=f(1)$ and let $\beta : F(\mathbb{R}) \to \mathbb{R}$ be defined by $\beta(f)=f(2)$ Prove that $\alpha$ and $\beta$ are ...
2
votes
1answer
56 views

Is the zero ideal $\{0_{M_2(\mathbb{R})}\}$ maximal in $M_{2}(\mathbb{R})$?

Consider the ring $M_2(\mathbb{R})$ of all $2 × 2$ matrices over $\mathbb{R}$. Is the zero ideal $\{0_{M_2(\mathbb{R})}\}$ maximal in $M_{2}(\mathbb{R})$? We know, in the ring $\mathbb{Z}$, ...
1
vote
1answer
29 views

Commutators of a symmetric group

I am trying to prove that the commutator subgroup of $S_n$ ($S_n$ is a symmetric group on $[n]$), $[S_n,S_n]$ consists solely of commutators $s_1^{-1}s_2^{-1}s_1s_2$ for some $s_1,s_2\in S_n$. Any ...
-2
votes
0answers
47 views

Whether the number $a$ is algebraic over $\mathbb{Q}$ [closed]

Is the number $a = \displaystyle \left( \frac{(1 + \sqrt[3]{7})^{\tfrac{7}{5}}}{(\sqrt[3]{7} - 7)^3 + 77} \right)^{13}$ algebraic? If so, is algebraic degree of $a$ bounded by $15$?
0
votes
1answer
32 views

is the image of a group inside its profinite completion normal? characteristic?

Let $G$ be a finitely generated, residually finite group and $\widehat{G}$ its profinite completion. Must the natural image of $G$ inside $\widehat{G}$ be a normal subgroup of $\widehat{G}$? Must it ...
0
votes
0answers
7 views

What does a basis of an affine module correspond to in a torsor?

An affine module, if and only if the module is free, by definition has one or more bases. My understanding is that an affine module over the ring $\mathbb Z$ can be converted into an equivalent ...
1
vote
0answers
43 views

Bourbaki - Algebra Chapter IV - Section 6, Exercise 9(b)

Let $S_i(X_1,\dots,X_n)$ be the elementary symmetric functions in the variables $X_1,\dots,X_n$. Let $r_1,r_2,\dots,r_n$ be $n$ rational functions in the $X_1,\dots,X_n$. Let $T$ be a variable ...
0
votes
0answers
22 views

Is the profinite completion of $SL_2(\widehat{\mathbb{Z}})$ isomorphic to $\widehat{SL_2(\mathbb{Z})}$?

Is the profinite completion of $SL_2(\widehat{\mathbb{Z}})$ isomorphic to $\widehat{SL_2(\mathbb{Z})}$ ? Does anything change if we replace $SL$ with $GL$?