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.

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If $R$ is a commutative ring, the nilpotents form necessarily an ideal of $R$? [duplicate]

This is an algebra question from an exam a few years ago: Let $R$ be a ring, and let $N = \{a \in R: a^n = 0 \text{ for some } n \in \mathbb{N}, (n \text{ depends on } a) \}$. Prove or disprove: ...
-1
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1answer
19 views

I have to show it is isomorphic to $K = GF(p^{kd})$ [closed]

Suppose $F = GF(p^k)$ is a finite field. I know $F[C]$ is a field extension of $F$ with degree $d = \deg m$, and I have to show it is isomorphic to $K = GF(p^{kd})$ (where $C$ is a companion matrix ...
2
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2answers
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Can one decompose any ring into a (possibly infinite) product of indecomposable rings?

Let $R$ be a ring. Do there exist indecomposable rings $R_i$, $i\in I$, such that $R={\prod}_{i\in I} R_i$?
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2answers
26 views

Fixed elements of an automorphism [closed]

If $u$ is an automorphism of a field $K$, the elements of $K$ fixed by $u$ form a subfield. How do you prove this?
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1answer
295 views

Universal Mapping Property of Free Abelian Groups

Let S be a set and $F=F_S$ the free group on S. Let $F'$ be the commutator subgroup of $F$. Set $A=A_S = F/F'$, and call it the free Abelian group on $S$. Prove the universal mapping property of the ...
-1
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0answers
25 views

Direct sum of algebras [closed]

I was wondering if we have an algebra $$A=F_p[h_1, …, h_n] \oplus F_p[h_1, …, h_n] g$$ for all $p$, can we have the direct sum over Z? Thanks
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1answer
384 views

Classify relations “is greater than or equal to”

Classify the following relations as reflexive, irreflexive, symmetric, antisymmetric or transitive. Explain each property in the context of the question. “is greater than or equal to” on the set of ...
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1answer
38 views

Show that a Set S is a subring of $R \times R$

Question: Prove that$$S=\{ (r,r) | r \in R\}$$ is a subring of $R \times R$. Attempt: Proof: Let $(a,b)$ and $(c,d)$ $\in R$. As $(a,b) \cdot (c,d) = (ac,bd) \in S$. $(a-c, b-d) \in S$. I show ...
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1answer
19 views

What is a relation (finitely related module)?

https://en.wikipedia.org/wiki/Finitely-generated_module#Finitely_presented.2C_finitely_related.2C_and_coherent_modules I've understood the first part of the definition. Then, "M is isomorphic to ...
2
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1answer
84 views

Is a nontrivial finite group of order $n$ always isomorphic to a subgroup of $GL_{n-1}(\mathbb{Z})$?

I saw this question on an old qualifying exam: Let $G$ be a group of order $n\ge2$. Is such a group always isomorphic to a subgroup of $GL_{n-1}(\mathbb{Z})$? A simpler problem would be to show ...
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0answers
28 views

Intermediate “prime” extensions [Rotman]

Problem Assume $F$ contains the $k$th roots of unity, and let $R=F(\alpha)$, where $\alpha$ is a root of $x^k-a$ for some $a\in F$. Prove that there exist intermediate fields $$F=K_0\subset ...
3
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1answer
57 views

Isomorphic or equal?

Let $\sim_n$ be the usual equivalence relation of congruence modulo $n$ in $\mathbb{Z}$, i.e., for $a,b\in\mathbb{Z}$, $a\sim_nb\Leftrightarrow a-b=k\cdot n$ for some $k\in\mathbb{Z}$. For $n=0$ the ...
1
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1answer
62 views

Structure of the unit group $(\mathbb{Z}[i]/8\mathbb{Z}[i])^\times$

I know that $\mathbb{Z}[i]/8\mathbb{Z}[i]=\{a+ib \mid a,b\in\mathbb{Z}_8\}$. But I'm not able to comprehend what $(\mathbb{Z}[i]/8\mathbb{Z}[i])^\times$ is. Can someone please help me get its ...
2
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1answer
103 views

How to prove the global dimension of the polynomial ring $F[x_1,…,x_n]$ is $n$?

I am trying to prove that the global dimension of the polynomial ring $F[x_1,\dots,x_n]$, where $F$ is a field , is exactly $n$. By Koszul complex, I know its global dimension is greater than or ...
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4answers
211 views

Global dimension of free algebra.

Is there any easy way to see the global dimension of a free algebra $$ A=k\langle x_{1},\dots,x_{n} \rangle $$ is 1?
2
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1answer
123 views

Global dimension.

What is the global dimension of $\mathbb{Z}_{(p)}$ and $\mathbb{Z}_{(p)}/t\mathbb{Z}_{(p)}$, where $\mathbb{Z}_{(p)}$ is the local ring, $p$ prime and $p \mid t$? What is the global dimension of ...
25
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1answer
306 views

Group $G$ whose center $Z(G)$ is cyclic and with $G/Z(G)$ commutative

I have some issue to solve following exercise. The exercise is from a French book on Algebra (cours d'Algèbre) from Jean Querré. The book is from the 1970's. If the center $Z(G)$ of a group $G$ is ...
2
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1answer
39 views

Global dimension of an intermediate ring

Assume $A \subseteq B \subseteq C$ are noetherian integral domains, where $A$ and $C$ have the same finite global dimension $n$. Also assume that $C$ is a finitely generated $B$-algebra and $B$ is a ...
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2answers
118 views

Difference between centralizer and center groups?

This is probably stupid question, but I can't see the difference between the two subgroups: $$C_G(A)=\{g\in G| gag^{-1}=a,\forall a\in A\}$$ $$Z(G)=\{g\in G| ga=ag,\forall a\in G\}$$ Is the ...
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1answer
28 views

Number of Cosets of Intersection of Subgroups

Similar question has been asked on SE before but the problem statement is usually more specific and gives more information (in particular, tells you what to prove), but this problem asks to prove or ...
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1answer
43 views

$k[X_1,\ldots X_n]/(X_1)\cong k[X_2,\ldots, X_n]$

I'm trying to prove that $k[X_1,\ldots X_n]/(X_1)\cong k[X_2,\ldots, X_n]$, in order to do so, I'm trying to find an epimorphism such that $(X_1)$ is kernel, any suggestions? Thanks
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3answers
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Can you learn linear algebra with an abstract algebra book?

I am trying to learn both linear and abstract algebra. I already took a basic Matrix Theory course using Anton's linear algebra book (not very rigorous) and I want to self study the rest of the ...
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1answer
45 views

Difference between $(N+P)/N$ and $P/N$

If $N$ and $P$ are submodules of the $A$-module $M$ (where $A$ is a commutative ring with unity), why is there a difference between $(N+P)/N$ and $P/N$? If $x\in (N+P)/N$ then $x=n+p+N=p+N$ for some ...
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3answers
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How to find the Galois group of a polynomial?

I've been learning about Galois theory recently on my own, and I've been trying to solve tests from my university. Even though I understand all the theorems, I seem to be having some trouble with the ...
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0answers
16 views

Number of homomorphisms between finitely generated abelian group and a finite cyclic group

This is the situation: Suppose we have a finitely generated group $G= \mathbb{Z}^r \times E$ with $E$ it's tortion subgroup and $m=$ exponent of $E$ i.e. the least natural number such that $x^m=1$ ...
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0answers
38 views

Prime ideal in indecomposable commutative ring [closed]

Let $R$ be a commutative indecomposable ring with Jacobson radical $J$. When can we find a prime ideal contained in $J$?
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2answers
76 views

Semilocal commutative ring with two or three maximal ideals

Is there any equivalence condition for a commutative ring to have exactly two or three maximal ideals?
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5answers
874 views

How to solve a cyclic quintic in radicals?

Galois theory tells us that $\frac{z^{11}-1}{z-1} = z^{10} + z^9 + z^8 + z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z + 1$ can be solved in radicals because its group is solvable. Actually performing the ...
3
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2answers
458 views

Let G be a finite group with more than one element. Show that G has an element of prime order

Let $G$ be a finite group with more than one element. Show that $G$ has an element of prime order
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1answer
41 views

Why is $x^{100} = 1 \mod 1000$ if $x < 1000$ and $\gcd (x,1000) = 1$?

Let $U(1000) =$ the multiplicative group of all integers less than and relative prime to $1000$. "Show that for every $x \in U(1000)$ it is true that $x^{100} = 1 \mod 1000$." Been thinking ...
22
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1answer
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$x^p-c$ has no root in a field $F$ if and only if $x^p-c$ is irreducible?

Hungerford's book of algebra has exercise $6$ chapter $3$ section $6$ [Probably impossible with the tools at hand.]: Let $p \in \mathbb{Z}$ be a prime; let $F$ be a field and let $c \in F$. Then ...
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0answers
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construction field $F_n$ [duplicate]

I know that if $F_n$ is a finite field, then $n$ should be a prime power. I want some sources in order to learn how I can construct finite fields $F_n$ for such an $n$.
2
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1answer
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Need a hint to get started on this algebra problem.

The problem is: If it is known that a group G with m generators and exponent n is finite, can one conclude that the order of G is bounded by some constant depending only on m and n? Equivalently, are ...
5
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2answers
207 views

Dimension of $\operatorname{Hom}(V, W)$

What is the dimension of $\operatorname{Hom}(V, W)$ if at least one of the two vector spaces $V, W$ is infinite dimensional? In the sense of cardinal numbers. Thanks
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2answers
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Prove that $M$ is finitely generated and it is a semi-simple module.

Let $M$ be a $R$-module. It is given that intersection of all maximal sub-modules of $M$ is the zero module. Moreover the module is given to be Artinian. Prove that $M$ is finitely generated ...
4
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3answers
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Must the centralizer of an element of a group be abelian?

Must the centralizer of an element of a group be abelian? I see that the definition of centralizer is: Let $a$ be a fixed element of a group $G$. The centralizer of $a$ in $G$, $C(a)$, is the ...
2
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1answer
55 views

Can't we really add together two points on a manifold?

Let us consider a classical mechanical system with observables being smooth functions $C^\infty(X)$ on a Poisson manifold $X$. The algebra of observables will be denoted as $A$ Next we can define ...
2
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3answers
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Prove that if $M$ is finitely generated then it is Artinian.

Let $M $ be a semisimple $R$-module. Prove that if $M$ is finitely generated then it is Artinian. To show this we have to prove that every non-empty collection of sub-modules of $M$ has a minimal ...
8
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2answers
78 views

$n-1$ dimensional permutation module for $S_n$

Say $n \ge 5$. Let $P$ be the $(n-1)$ dimensional permutation module for $S_n$, i.e. the permutation representation on $\{(x_1, \dots, x_n) \in {\bf C}^n: \sum x_i = 0\}$. Prove that: $\wedge^2P$ ...
3
votes
1answer
42 views

A ring isomorphic to its square [duplicate]

Is there an example of a ring $A$ (with unity) which is isomorphic as unital rings to $A\times A$? Any such ring can't have invariant basis number so in particular can't be commutative.
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1answer
31 views

Trouble with a proof: $(p^n - 1 , e)=1$ for $e\in \mathbb{N}$, p prime

I'm having trouble understanding a proof. The Lemma states: For every natural number $e$ there are infinitely many prime powers $q$ with $(q-1,e)=1$. The prove is as follows: Write $e=2^km$, m odd. ...
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Transitivity of discriminant for flat algebras

Let $A$ be an finite flat $R$-algebra and $A'$ be an finite flat $A$-algebra such that it is also finite flat as an $R$-algebra. Then we have a notion of discriminant ideals ...
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1answer
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Prove that $f$ is a nonzerodivisor on $R[x_1,\dots,x_r]/IR[x_1,\dots,x_r]$ for every ideal $I$ in $R$

Let $R$ be a Noetherian commutative ring with unity, and $S=R[x_1,\dots,x_r]$. Let $f\in S$ be a nonzerodivisor of $S$. Suppose that the ideal generated by the coefficients of $f$ is $R$. How to ...
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1answer
50 views

Addition of points on elliptic curves over a finite field

I have found the following formulas for the coordinates of $P+Q$ given that $P = (x_{1}, y_{1})$ and $Q = (x_{2}, y_{2})$ are points on a general curve $y^2 = x^3 + ax + b$ over $\mathbb{R}$: $$P + Q ...
0
votes
1answer
59 views

Solvability of transitive group

Let $G$ be a transitive subgroup of $S_p$ where $p$ is an odd prime number. Now consider the following assumptions - $(i)$ $G$ is solvable. $(ii)$ If $\sigma \in G$ and there exist $h\ne j$ such ...
11
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1answer
173 views

To find continuous functions on $\mathbb R$ which preserve certain algebraic structures

Can we determine all non-constant continuous functions $f:\mathbb R \to \mathbb R$ such that for every subgroup $G$ of $(\mathbb R,+)$, $f(G)$ is also a subgroup of $(\mathbb R,+) $ ? And ...
2
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1answer
63 views

Completion of the proof of theorem 3.3 in Dale Husemoller: Elliptic Curves

I want to read the proof of the following theorem: This is from p.35. But it is not complete there. There is written that: Can someone tell me where I can find the rest of the proof? Any other ...
3
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2answers
36 views

Prove image of symmetric group into additive group of real numbers is zero

Suppose $ f : S_{n} \rightarrow (\mathbb{R} , + , 0 , - )$ is a group homomorphism. Prove $ f(S_{n}) = {0} $, i.e., $f(\sigma) = 0$ for every $ \sigma \in S_{n}$with $n \geq 1 $ I cannot seem to find ...
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2answers
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Quotients of Solvable Groups are Solvable

I just proved that subgroups of solvable groups are solvable. So given that $G$ is solvable there is $1=G_0 \unrhd G_1 \unrhd \cdots \unrhd G_s=G$ where $G_{i+1}/G_i$ is abelian and for $N$ a normal ...
3
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3answers
74 views

Finding the number of elements of particular order in the symmetric group

I know how to find the order of element in any group $G$, for example the order of $2$ in $\mathbb{Z}_5$ is $5$ as $2 + 2 + 2 + 2 + 2 = [10]_5 = 0 0$, which is the identity in $\mathbb{Z}_5$. But, ...