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)

1
vote
2answers
63 views

Show that there is odd number of elements of a finite group satisfying $x^3=e$

Show that: Show that there is odd number of elements of a finite group satisfying $x^3=e $? And even number of elements satisfying $x^2\neq e$??? I donot have any idea how to start.
0
votes
0answers
21 views

Complex extension isomorphic to $\mathbb{R}$?

Let $K$ be some field extension of $\mathbb{Q}$ containing some complex number $c=a+bi$ with $a,b\in \mathbb{R}$ and $b\neq 0$. Is it possible that $K\cong \mathbb{R}$ as fields? I tried to disprove ...
-1
votes
0answers
13 views

Describe the prime ideals of Ring $R$ in terms of their generators. [duplicate]

Let $R:=\Bbb C[x,y]$ denote the ring of polynomials in the variables $x$ and $y$, with complex coefficients. Describe the prime ideals of $R$ in terms of their generators. Prime ideals are ideals ...
-1
votes
1answer
40 views

SHow that $\mathbb{Q}(i,\sqrt{3}) = \mathbb{Q}(i+\sqrt{3})$ [on hold]

I am stuck on how to solve this problem, any hints on what would be useful here?
1
vote
2answers
32 views

Is the kernel of the induced homomorphism of a group action of G on a subgroup H a subgroup of H?

If we have a subgroup H of a group G, and act by left regular action on the set A of left cosets of H in G. This induces a homomorphism $\phi: G \rightarrow S_{n}$, where ker($\phi$) is a normal ...
1
vote
3answers
52 views

Show that two rings are not isomorphic

I don't know how to show (or why) $M_{2\times2}\mathbb{(R)}$ is not isomorphic to $\mathbb{R}[x]/(x^4-1)$ does it have something to do with the order of coset representatives of the quotient group? ...
0
votes
1answer
23 views

Order $4$ subgroup of alternating group $A_4$

I ran into the following problem: Let $H$ be the subgroup $H = \{e, (1\,2)(3\,4), (1\,3)(2\,4), (1\,4)(2\,3)\}$ in $G = A_4 = H \cup \{(1\, 2\, 3), (1\, 3\, 2), (1\, 2\, 4), (1\,4\,2), ...
2
votes
1answer
27 views

Show that every polynomial of degree $1,2,$ or $4$ in $\mathbb{Z}_2[x]$ has a root in $\mathbb{Z}_2[x]/(x^4+x+1)$.

The problem: Show that every polynomial of degree $1,2,$ or $4$ in $\mathbb{Z}_2[x]$ has a root in $\mathbb{Z}_2[x]/(x^4+x+1)$. My attempt: I know that the polynomials $x$ and $x+1$ have ...
0
votes
2answers
29 views

If $a$ and $b$ are nonzero integers such that each is a divisor of the other, show that $a = ± b$ .

I tried many approaches to this problem. I believe that if I did $b|a=m$ and $a|b=n$ and set $m=n$, then $a$ and $b$ would be equal. Is that how it should be done? If not, please help me out. Thanks.
3
votes
0answers
46 views

What do you call the category of groups with surjections (injections)?

Take the category $Grp$ and throw out all morphisms that are not surjections. I think what's left is still a category. The composition of two surjections is surjective, and identities exist ...
2
votes
1answer
20 views

Torsion in module over the ring of convergent power series

I need to understand a passage from a paper which I don't quite understand. Let $M$ be a module over the ring $\mathbb C\{t\}$ of convergent power series. We want to show that $M$ is torsion-free, ...
2
votes
1answer
64 views

Is $\operatorname{Mat}(2,\mathbb{R})$ isomorphic to $\operatorname{Mat}(3,\mathbb{R})$? [on hold]

As the title. Is $\operatorname{Mat}(2,\mathbb{R})$ isomorphic to $\operatorname{Mat}(3,\mathbb{R})$? I cannot seem to find any bijection that will do. Thanks. $\operatorname{Mat}(n,\mathbb{R})$ ...
0
votes
1answer
29 views

Define a new addition ⊕ and multiplication on Z by a⊕b = a + b−1 and ab = a + b−ab.

a+b and ab are the usual integer addition and multiplication. You can assume that this new operation forms a ring, say R is the set of integers with these operations. Then does R have zero-divisors? ...
0
votes
0answers
39 views

Isomorphism theorems for topological groups

I know that the second isomorphism theorem for groups doesn't hold for topological groups, the version that I have for the second isomorphism theorem is: If $G$ is a group, $H$ a subgroup of $G$ and ...
1
vote
0answers
46 views

What is the purpose of homomorphisms?

I know that a mapping $\phi:A\to B$ is a homomorphism provided that $$\phi(A*B)=\phi(A)\times\phi(B)$$ where $*$ and $\times$ are two operators on the algebraic structures $A$ and $B$ respectively. In ...
-2
votes
1answer
29 views

Algebra and field irreducibility [on hold]

I am having trouble with a series of algebra questions on fields. Let $f$ in $F[x]$ be an irreducible polynomial with coefficients in a field $F$ and of degree $\geq 1$. Let $L = F[x]/(f(x))$. ...
2
votes
1answer
29 views

Algebra, finding the elements of the field and solving irreducible polynomials

I'm trying to do this problem from a practice final but there are no solutions. I honestly am pretty stumped. My thought was since it has 7 elements, then the degree of the polynomial must be one ...
0
votes
0answers
34 views

Determine the group of units of a subset of $M_n(\mathbb{C})$

Let $R$ be a commutative ring. Let $R=\bigg\{\begin{bmatrix}u & v\\ 0 & u\end{bmatrix}:u,v\in\mathbb{C}\bigg\}$. Determine the group of units $R^{\times}$ of $R$. My try: Let ...
2
votes
0answers
28 views

Bezout relation with integral coefficients

Suppose I have two monic polynomials $f$ and $g$ with coefficients in $\mathbb{Z}$. I also suppose that $f$ and $g$ are coprime as polynomials over $\mathbb{Q}$. In particular, there exists a Bezout ...
3
votes
1answer
53 views

Why is $R((X))$ defined as follows?

Let $R$ be a commutative ring. Then $R((X))$ is defined as the set of all $\sum_{n\geq N} a_n X_n$ where $N\in\mathbb{Z}$ and is called "The Formal Laurent series". But why? Why don't we consider ...
-3
votes
0answers
30 views

Let G be a group, and suppose that H is a subgroup of G of order 5, and no other subgroup of G has order 5. Show that H is a normal subgroup of G. [on hold]

Let G be a group, and suppose that H is a subgroup of G of order 5, and no other subgroup of G has order 5. Show that H is a normal subgroup of G. Picture of my proof I got 5/10 on this problem. It ...
-4
votes
0answers
32 views

need help with abstract algebra [on hold]

Let $A_4 \le S_4$, and $$A_4 = \{ (1), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3), (1 2 3), (1 3 2), (1 2 4), (1 4 2), (1 3 4), (1 4 3), (2 3 4), (2 4 3) \}$$ Find the conjugacy classes and the class ...
-3
votes
0answers
62 views

Using Gap system. [on hold]

I'm new at the GAP. Probably I can't use this system, what I type doesn't work. For instance why the following doesn't work? for i in [1..1160] do Print("Processing semigroup number ",i,"\n"); ...
1
vote
0answers
41 views

Does Nagata theorem hold in a field that is not algebraically closed?

Let $R$ be a finitely generated $k$ - algebra and $G$ be a reductive group acting rationally on it. Then a theorem of Nagata says that the invariant ring $R^G$ is also finitely generated. Here $k$ ...
0
votes
0answers
7 views

for every left ideal $L$ of $R$ and $R$-module homomorphism $g:L\to A$, there exists $a\in A$ such that $g(r)=ra$ for every $r\in R$.

Let be a $R$ ring with a identity. An $R$-module $A$ is injective iff for every left ideal $L$ of $R$ and $R$-module homomorphism $g:L\to A$, there exists $a\in A$ such that $g(r)=ra$ for every ...
0
votes
1answer
16 views

Center of finite dimensional division $\mathbb{R}$-algebra?

Let $D$ be a finite dimensional division $\mathbb{R}$-algebra. Why is it that $Z(D)=\mathbb{R}$ or $Z(D)=\mathbb{C}$? I have seen an explanation: It is because $\mathbb{C}$ is the only non ...
0
votes
0answers
9 views

Endomorphism of Central Simple Algebra

Let $A\in\mathscr{C}(F)$, i.e. $A$ is a central simple algebra. Show that $\text{End}_F(A)\cong M_n(F)$ as $F$-algebras where $n=\dim A$. My idea is to consider $A$ as a $F$-vector space, then ...
2
votes
1answer
53 views

About the $1$ of ring

I could not find neither a proof nor a counterexample, can anyone solve this? Let $A$ be a finite dimensional $k$-algebra. (It not necessarily has $1$.) If $$\mu:A\otimes A \rightarrow A,\ ...
0
votes
2answers
45 views

Subfield of a finite field

I have started studying field theory and i have a question.somewhere i saw that a finite field with $p^m $ elements has a subfield of order $p^m $ where $m$ is a divisor of $n $.My question that if ...
1
vote
1answer
21 views

Isomorphism between the group $(\mathbb Z[x], +)$ and $(\mathbb Q_{>0}, .)$ [duplicate]

In one of my assignment, I was told that $(\mathbb Z[x], +)$ and $(\mathbb Q_{>0}, .)$ are isomorphic, with $$\phi (\sum_{k=0}^n a_k x^k) = \prod_{k=0}^n p_k^{a_k}$$ is a one-to-one surjective ...
-2
votes
1answer
49 views

Subgroups of $\mathbb{Q}/ \mathbb{Z}$ [on hold]

I was asked to show in my exam that all the subgroups of $\mathbb{Q}/ \mathbb{Z}$ are of the form $\{\mathbb Z + a/p^i\}$ where $p$ is a prime number and $0\leq a \lt p^i$, where $i$ varies over all ...
0
votes
1answer
18 views

Confused about order in Opposite Algebra

I am facing confusion in the "order of multiplication" regarding the opposite algebra $B^o$ in the following working: Define a right $B^o$-module $M_\phi$ where $M_\phi=M$ via $m\cdot b=m\phi(b)$. ...
0
votes
1answer
15 views

Problem on field extension related to irreducible polynomial

Suppose $\gamma,\gamma'\in\Bbb C$ are distinct roots of the same irreducible polynomial $p\in\Bbb Q[x]$. Suppose $x^2-5$ is irreducible in $\Bbb Q(\gamma)[x] $. Show that it is also irreducible in ...
-3
votes
0answers
33 views

if $\phi : G \mapsto G'$ $y \mapsto \left[ x,y \right]$ why $\phi$ is onto? [on hold]

if $\phi : G \mapsto G'$ $y \mapsto \left[ x,y \right]$ why $\phi$ is onto?
-5
votes
1answer
30 views

abstract algebra [on hold]

Show that: the multiplicative groups $\mathbb{R}{}$ and $\mathbb{C}{}$ (both excluding {0}) are not isomorphic the additive groups $\mathbb{R}$ and $\mathbb{Q}$ are not isomorphic the groups $D_4$ ...
0
votes
2answers
23 views

Abstract Algebra- Factor Group Generator Question

So far I have that the factor group is equal to: {(2,1), (1,2), (0,0)}, {(0,1), (2,2), (1,0)}, {(0,2), (2,0), (1,1)}. However, I'm having trouble finding, even understanding what a generator of a ...
-1
votes
2answers
30 views

Proof in modern algebra. Prove $(n-1)! = -1 \ (\textrm{mod n})$ iff n is prime [duplicate]

Prove $(n-1)! = -1 \ (\textrm{mod n})$ iff n is prime I can understand how the first part of the proof $(n-1)!=-1 \ (\textrm{mod n})$ is true if n is prime simply by testing it out. However, I'm ...
1
vote
1answer
26 views

Multiplicative inverse of $x+f(x)$ in $\Bbb Q[x]/(f(x))$

So I have $f(x) = x^3-2$ and I have to find the multiplicative inverse of $x + f(x)$ in $\mathbb{Q}[x]/(f(x))$. I'm slightly confused as to how to represent $x + (f(x))$ in $\mathbb{Q}[x]/(f(x))$. ...
0
votes
1answer
33 views

Show that $\pi(Z)$ acts as a scalar over $\mathbb{g}$

Let $(\pi, V)$ be a finite dimensional irreducible representation of $\mathbb{g}$ $V$ is a vector space of homogeneous polynomials in 3 variables of degree d over $\mathbb{R}$ ...
-7
votes
0answers
17 views

I want solutions pdf of A.joseph gallion book abstract algebra [on hold]

I find it all over internet .if anybody have this book please share this solutins of gallion book on modern algebra.
0
votes
0answers
14 views

Decompose $V \otimes V \otimes V$ into irreducible representations of $SL_2(\mathbb{R})$

Let $V=\mathbb{C^2}$ be the standard representation of $SL_2(\mathbb{R})$ Decompose $V \otimes V \otimes V$ into irreducible representations of $SL_2(\mathbb{R})$ I will just consider ...
0
votes
2answers
21 views

Decompose the representation $V$ of $SO_2$ into irreducible representations

Let $V=\mathbb{C^2}$ be the standard representation of $SO_2$ Decompose $V$ into irreducible representations The standard unit vectors of $\mathbb{C^2}$ are $e_1$ and $e_2$ I am not sure how ...
0
votes
1answer
17 views

prove the unity cannot map to the zero in a ring homomorphism? [on hold]

If $A$ and $K$ are nontrivial rings and $f: A \to K$ is an onto ring homomorphism then $f(1_A)\neq 0_K$. My idea was to try to use the kernel somehow, but I'm not sure how to show this.
-3
votes
1answer
35 views

Kernel of a homomorphism: $f(a)=f(x)$? [on hold]

Suppose $A$ and $K$ are rings with $f: A \to K$ a homomorphism. Prove that for any $x \in a + \ker(f)$ we have $f(x)=f(a)$. Im not sure how to start this, any help is appreciated!
0
votes
1answer
18 views

Ideal generated by given integers verification.

The question reads: Find the positive generator of the smallest ideal in $\mathbf Z$ containing the following ideals: a. $(4)$ and $(18)$. My answer is $(m)=(4)$. b. $(6)$ and $(35)$. My ...
-1
votes
2answers
30 views

if a is a unit of $A$, it is also a unit of the quotient ring? [on hold]

Suppose $A$ is a nontrivial commutative ring with unity and $S$ is an ideal of $A$ s.t. $S\ne\{0_A\}$ and $S\ne A$ . Prove or find a counterexample: if $a\in A$ is a unit in $A$, then $a + S$ is a ...
-2
votes
2answers
27 views

If I and J are isomorphic ideals of a ring R, does it follow that $R/I \simeq R/J$?

The title pretty much sums it up. We know that $R/I \simeq R/J$ does not necessarily imply $ I \simeq J$. But does the converse hold? I can't find any counterexample and all my efforts in proving it ...
-1
votes
1answer
32 views

13th root of 2 in field $\mathbb{F}_{13}$

Is there an easy to find $13^{th}$ root of $2$ in the field $\mathbb{F}_{13}$? I'm having trouble finding one here. Thanks!
-1
votes
1answer
14 views

About subgroups and factor groups of metabelian groups [on hold]

If $G$ is a metabelian group, show that: 1/ All its subgroups and and factor groups are metabelian; 2/ $G$ is metabelian if and only if its commutator group $G'$ is abelian. Thanks
0
votes
1answer
42 views

The ideal is maximal - Show the inclusion [on hold]

I want to show that the ideal $(x,y)$ is maximal in $F[x,y]$ and that it holds that $(x,y)^2\subseteq (x^2, xy, y^2)\subset (x^2, y)\subset (x,y)$. Knowing that the principal ideal $(x)$ is ...