Abstract algebra is a study of algebraic objects. Some of the more common algebraic objects are Groups, Rings, Fields, Vector spaces, Modules, and other advanced topics.
3
votes
2answers
60 views
Proving that $\langle{M}\rangle$ is a subgroup of $G$
I found a task suggestion in my lecture book that I unfortunately can't properly solve. It translates into:
$M$ is a non-empty subset of the group $(G,*)$. Prove that $\langle{M}\rangle$ is a ...
-1
votes
1answer
31 views
Determine a set of generators for operation of Ideals
$(f )\cap(g)$ where $f, g$ are relatively primes in a factorial domain.
I know that $(f)\cap(g)=(LCM(f,g))$. But I can't prove this correctly. Please help me in problem.
-1
votes
1answer
38 views
Splitting field questions.
$K$ is a field and $f \in K[x]$ with splitting field $L$. Show that $[L:K] \le n!$, where $n$ is the degree of $f$.
$f \in \mathbb{Q}[x]$ is a cubic polynomial and $K$ is its splitting field. What ...
0
votes
2answers
44 views
Field extension question.
F, K, L are fields. F is extended to K, and K is extended to L. Show that $[L:F]=[L:K] \cdot [K:F]$.
Also consider the extension from $\mathbb{Q}$ to $\mathbb{Q}(\alpha)$ where $\alpha = \sqrt{3} + ...
3
votes
0answers
43 views
GCD in a subring is GCD in a bigger ring
Let $R$ be a UFD which is a subring of an integral domain $S$. If $r_1$ and $r_2$ are two nonzero elements of $R$ with GCD $d$, is it true that $d$ is also a GCD of $r_1$ and $r_2$ in $S$?
I know ...
2
votes
1answer
95 views
Can we embed the ordinal and cardinal number systems into larger, more convenient systems of arithmetic?
We can embed $\mathbb{N}$ in a larger number system, such as $\mathbb{Z}$, $\mathbb{Q}$ or $\mathbb{R}$, for convenience. Now since $\mathbb{N}$ is extended by $\mathrm{Ord}$ and $\mathrm{Card}$, the ...
3
votes
1answer
24 views
Question regarding nilpotent ideals of a ring.
I am working on the following:
An ideal $N$ is called nilpotent if $N^n$ is the zero ideal for some $n\geq1$. Prove that the ideal $p\mathbb{Z}/p^m\mathbb{Z}$ is a nilpotent ideal in the ring ...
4
votes
2answers
64 views
How to prove these two groups are isomorphic?
Let $U_p$ be the subgroup of $\mathbb{C}^*$ satisfying that: for any $x\in U_p$, there exists an integer $n$ such that $x^{p^n}=1$.
Let $K$ be a algebraic closed field with characteristic $0$. Denote ...
1
vote
0answers
62 views
A counterexample for $\operatorname{Ass}(M_1+M_2)$ [duplicate]
$\newcommand{\Ass}{\operatorname{Ass}}$
Let $A$ be a Noetherian ring and let $M$ be an $A$-module. Suppose $M=M_{1}+M_{2}$, then we have $\Ass(M)\supset \Ass(M_{1})\cup \Ass(M_{2})$. What is an ...
2
votes
1answer
65 views
Relationship of factors of gcd [duplicate]
Assume $\text{gcd}(m,n) = {mx + ny} = d$
Prove that $\text{gcd}(x,y) = 1 = d'$
Now for the solution, we know that $\text d'|x$ , $\text d'|y$
so $\text d'|mx + ny$
so $\text d'|d$
After this? ...
0
votes
0answers
26 views
Normal endomorphism
I have a question about normal endomorphism. In class, we said that normal endomorphisms in finite dimensional real vector spaces are always of the form that we have some eigenvalues and further ...
3
votes
6answers
65 views
$\mathbb Z_{p^n}$ is not the direct product of any family of its proper subgroups
I'm trying to solve this question of Hungerford's Algebra book:
$S_3$ is not the direct product of any family of its proper subgroups.
The same is true of $\mathbb Z_{p^n}$
The first case is ...
6
votes
1answer
76 views
What are the ring morphisms $\mathbb{Q}[[X]]\to R$ for a ring $R$?
If $\mathbb{Q}[[X]]$ is the ring of power series over a field $F$, then can we describe ring morphisms $\mathbb{Q}[[X]] \to R$ for rings $R$ in simple terms?
I am guessing that a "substitution ...
0
votes
2answers
53 views
How many elements are in the kernel of the homomorphism $f:(\mathbb{Z}/154\mathbb{Z})^\star \to (\mathbb{Z}/154\mathbb{Z})^\star $ where $f(x)=x^5$?
How many elements are in the kernel of the homomorphism $f:(\mathbb{Z}/154\mathbb{Z})^* \to (\mathbb{Z}/154\mathbb{Z})^* $ where $f(x)=x^5$? The group operation in this case is multiplication with ...
2
votes
2answers
81 views
Prove that $SL_{2}(F_{3})$ can be generated by 2 elements
I'm doing an exercise in Dummit book "Abstract Algebra" and stuck on this problem. I'm so grateful if anyone can help me solve this. Thanks so much.
Prove that $SL_{2}(F_{3})$ is the subgroup ...
11
votes
4answers
257 views
Showing that $\sqrt \pi$ is transcendental
I'm trying to use the fact that $\pi$ is transcendental to show that $\sqrt \pi$ is also transcendental over $\Bbb{Q}$ . I don't know any theorems about algebraic and non-algebraic numbers so I don't ...
5
votes
5answers
113 views
Proving that additive groups are isomorphic $(n\mathbb{Z}$ and $m\mathbb{Z}$) or not ($\mathbb{Q}$ and $\mathbb{Z}$)
I switched my study course and am currently in the process of self-studying to catch up with the first few weeks of my linear algebra lecture which is why I'm still not really confident in how to ...
2
votes
2answers
60 views
Abstract proof simple group
Prove that a finite $p$-group $G$ is simple if and only if |$G$|=$p$.
We have $G$$\neq${$1$} and $G$ has no normal subgroups other than {$1$} and $G$ itself for it to be simple.
So if |$G$|=$p$ then ...
1
vote
1answer
58 views
Find all the equivalence classes of $ℜ$
Let $f:ℝ→ℝ$ be a real analytic function. Assume that $f$ has infinitely may zeros. We define the equivalence relation $ℜ$ on $G(f)$ via $(x_1,f(x_1))ℜ(x_2,f(x_2))$ if and only if $f(x_1)=f(x_2)$. Here ...
2
votes
3answers
75 views
Examples of epimorphisms which are not split epimorphisms?
Are there some examples of epimorphisms which are not split epimorphisms? Thank you very much.
3
votes
1answer
60 views
All the Associated Primes are minimal.
Let $R$ be a commutative Noetherian ring with unit and let $I$ be a fixed ideal. I am sorry if the following turns out to be a very silly question.
1) Suppose $\operatorname{Ass}(R/I)$ are all ...
2
votes
1answer
31 views
Abstract Cosets Dihedral
Consider $D_8$ = {$1,r,r^2,r^3, b,br,br^2,br^3$} and the subgroups $H$ = $<r^2>$ and $K$=$<b>$ of $D_8$.
List all the left cosets of $ H$ in $ D_8 $.
I have:
1$\cdot$$H$ = $r^2$$H$ = ...
3
votes
1answer
37 views
Dihedral groups and normal subgroups
Consider
$D_8$ ={$1,r, r^2, r^3,b,br,br^2,br^3$} and the subgroup, $H$={$1,r^2$} and $K$={$b$} of $D_8$
I really need some help with these particular problems.
Show that $H\lhd$$D_8$, but ...
-3
votes
7answers
352 views
Irrational roots don't exist [closed]
I'm going through Apostol's Calculus vol. 1.
It is excellent. But I was surprised to see him "prove" each number has a square root.
Inverse functions like divide usually introduce some invalid ...
1
vote
1answer
41 views
What is a “distinguished automorphism” of a field?
Math people:
The title is the question. The reason I am asking is that I am trying to determine exactly what fields can be used for an inner product. I posed that question at ...
1
vote
1answer
29 views
about the fraction of a ring
In chapter 6 of Undergraduate commutative algebra, there is a question asking when given a ring $A$ and a fixed multiplicative set $S$,there is a maximal multiplicative set $T$ such that ...
7
votes
0answers
101 views
Find all maximal subrings of $\mathbb{C}[x]$
Definition: A maximal subring $S$ of $R$ is a subring such that if $S \subseteq T \subseteq R$ then $T=S$ or $T=R$.
Find all maximal subrings of $\mathbb{C}[x]$.
Clearly $\mathbb{C}[x^2,x^3]$ ...
1
vote
1answer
42 views
normal subgroups: proving $H\lhd D_8$, given $K \ntriangleleft D_8$, knowing $KH$
I was asked to prove $H\lhd$$D_8$, given that K$\ntriangleleft$$D_8$. But later in the same problem I am suppose to prove that $HK\lhd$$D_8$.
I thought that if $H\lhd$$D_8$ and $K\lhd$$D_8$ then ...
4
votes
0answers
40 views
Main use of tensor, symmetric and exterior algebras outside differential geometry?
So I've seen these defined when constructing differential forms and in the construction of integration of manifolds. However, these seem to be a standard subject in most graduate algebra books, yet, ...
2
votes
2answers
46 views
Order of Subgroup , and exponentiating elements
I understand how to compute the orders but can someone explain why $ x $ = $ [1] $ goes to $ x^2 $ = $ [2] $
14
votes
12answers
528 views
Proving that $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ are not isomorphic.
I am trying to prove that $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ are not isomorphic. I was thinking of arguing the following:
Suppose there exists an isomorphism $\varphi: ...
3
votes
2answers
41 views
Order of the unit group of a finite field F if for all two subgroups of F one is contained in the other.
Let $F$ be a finite field. Prove that the following are equivalent:
i) $A \subset B$ or $B \subset A$ for each two subgroups $A,B$ of $F^*$.
ii) $\#F^*$ equals 2, 9, a Fermat-prime or $\#F^* -1$ ...
0
votes
1answer
78 views
$(U\circ T)^{*} = T^{*}\circ U^{*}$
Let $T : V \longrightarrow W$ and $U : W \longrightarrow Z$ be linear maps. How do I prove that $(U\circ T)^{*} = T^{*}\circ U^{*}$? I'm used to seeing $V^{*}$ not $(U\circ T)^{*}$. Any help is ...
1
vote
1answer
27 views
Algebra generator's
Consider the comutative ring with identity $R=\Bbb R[x,y]$ of polynomials in two variables $x,y$ with coeficients in the real numbers. Let $I_p$ denote the ideal generated in $R$ by $p(x,y)=x+3y^2$. ...
2
votes
4answers
29 views
Irreducibility of a polynomial over rationals.
I am given the polynomial $x^4+1$ and I am asked to prove that it is irreducible in $\mathbb Q[x]$. I was just wondering if it is enough to show that $x^4+1$ does not contain a root in $\mathbb Q$ and ...
0
votes
0answers
41 views
What is the proper name of this class of rings?
There is a property of rings that seems to be quite natural, but I can't seem to find a short name for it. A commutative ring with unit $R$ has this property if and only if it has no divisors of $0$, ...
2
votes
0answers
39 views
p-sylow-subgroups
I have to show that given a normal subgroup $N$ of $G$, $|N|=p^k$ for some $k\in \mathbb{N}$, that $N\subseteq P$ where $P$ is a $p$-sylow-subgroup of $G$, for all $P$ $p$-sylow-subgroups. What I have ...
6
votes
1answer
42 views
Square roots of quaternions
In class we saw that $-1$ has infinitely many square roots in the ring of quaternions. Is it possible to compute the square roots of a given nonreal quaternion? What do they look like?
3
votes
1answer
28 views
Factorization of polynomial over arbitry commutative ring
I have a somewhat silly sounding question: let $R$ be an arbitrary commutative ring with $1$. Let $f \in R[X]$.
Do
a) $f(1) = 0 \Rightarrow \exists g \in R[X]: f= (X-1)g$
b) for some $a \in R$: ...
3
votes
1answer
18 views
multiplicity of irreducible components of S3 modules
Let V denote the 2 dimensional irreducible standard module for $S_3$. I want to find multiplicity of each of irreducible components of $V^{\otimes ^{10}}$ , by writing the character for $V^{\otimes ...
6
votes
3answers
83 views
Largest order of an element in a group
How to determine the largest order of an element in the group $\mathbb Z_4 \times \mathbb Z_{18} \times \mathbb Z_{15}$.
I know the order of the element $(a, b)$ in the direct product $\mathbb ...
1
vote
1answer
60 views
Localization of $K[x,y|x^2-y^3]$ and $K[x,y|xy]$ at $\langle x,y\rangle$ and $\{\text{non-zero-divisors}\}$ (exercise in SICA)
In Greuel & Pfister's A Singular Introduction to Commutative Algebra, p. 38, there is written:
So we have rings
$$\begin{array}{l l}
R_1:= K[x,y|x^2\!-\!y^3], & R_4:= K[x,y|xy],\\
R_2:= ...
0
votes
2answers
45 views
Subgroups of $\mathbb{Z}^n$ of rank $n$
I found from here that groups $A \supset B \supset C$ with $A \simeq C$ does not imply $A\simeq B$. How about the case of $A\simeq \mathbb{Z}^n$? Is it still false?
4
votes
0answers
25 views
Relationship between representations of $\mathfrak{sl}_{2n}\mathbb{C}$ and $\mathfrak{sp}_{2n}\mathbb{C}$
If $V=\mathbb{C}^{2n}$ denotes the standard representation of $\mathfrak{sl}_{2n}\mathbb{C}$, what can we say about $\wedge^kV$ in terms of the standard representation $W$ of ...
2
votes
2answers
55 views
Why does $\sum_{k=n_0}^n{P(k)}$ is a polynomial in $n$ of degree $d$?
Let $P(X)$ be a polynomial over $\mathbb{Z}$ of degree $d-1$ and $n_0$ be some constant positive integer. Then why does $\sum_{k=n_0}^n{P(k)}$ is a polynomial in $n$ of degree $d$?
1
vote
1answer
23 views
Operation of $O_2$ on the plane.
I am currently reading through M. Artin's Algebra 2.ed. In chapter 6, remarks similar to the following are often made with little explanation:
"Unless an origin is chosen, the orthogonal group $O_2$ ...
1
vote
2answers
98 views
Computing $\mathrm{Hom}(\mathbb Z,\mathbb Z)$ as $\mathbb Z$-module
My algebra is weak I need help computing $\mathrm{Hom}(\mathbb Z_n,\mathbb Z)$, $\mathrm{Hom}(\mathbb Z_n,\mathbb Z_m)$ and also $\mathrm{Hom}(\mathbb Z,\mathbb Z)$ as $\mathbb Z$-modules. Also books ...
1
vote
1answer
34 views
Image of the projection map onto an irreducible module
Let $A$ be a $K$-algebra $V$ be an $A$-module.Let $V=\bigoplus W_j$ with $W_j$ irreducible for all j .Let $M$ be an irreducible $A$-module. And $N=\Sigma \{W_i: W_i$ is isomorphic to $M\}$. Now let ...
3
votes
2answers
56 views
Groups and matrices
Let $K$ be the additive group of $\mathbb Z\oplus \mathbb Z$. If $A = \left(\begin{array}{cc}
a & b\\
c & d
\end{array}\right)$ is an $2\times 2$ matrix where $a, b, c, d$, are in $\mathbb Z$, ...
5
votes
2answers
105 views
If $f(x)$ is an irreducible polynomial of degree n, then the cardinality of its Galois group is divisible by $n$.
If $f(x)$ is an irreducible polynomial of degree $n$, then the cardinality of its Galois group is divisible by $n$.
I know I need to use the Tower Theorem, but I can't figure out how to get from ...
