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.
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0answers
19 views
Synthetic Division with mods
$x^4+x+1$/
$2x^2$+1
In $F_5$ (means mod 5)
I said let the leading coefficient be 2.
Since $3$ $*$ $2$ - $5$ $*$ $1$ $=$ $1$, choose 3 to multiply
$3$($2x^2$ $+$ $1$)= $x^2$ + $3$ (this is ...
1
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2answers
63 views
Newbie vector spaces question
So browsing the tasks our prof gave us to test our skills before the June finals, I've encountered something like this:
"Prove that the kernel and image are subspaces of the space V: $\ker(f) < V, ...
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0answers
48 views
Theory of rings $1$ [closed]
$D$ is a division ring. All the prime ideals and semiprime the $A=T_3(D)$, the ring of upper triangular 3x3 matrix with coefficients in $A$. What would be the ideal?
2
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2answers
35 views
How to compute $\mathbb{Z}_{4} \times\mathbb{Z}_{2} \times \mathbb{Z}_{8}/ \langle(2,1,2)\rangle$?
I am trying to compute the factor group $\mathbb{Z}_{4} \times\mathbb{Z}_{2} \times \mathbb{Z}_{8}/ \langle(2,1,2)\rangle$. Here is what i did:
First i expand $\langle(2,1,2)\rangle$, it is equal ...
4
votes
3answers
77 views
on the commutator subgroup of a special group
Let $G'$ be the commutator subgroup of a group $G$ and $G^*=\langle g^{-1}\alpha(g)\mid g\in G, \alpha\in Aut(G)\rangle$.
We know that always $G'\leq G^*$.
It is clear that if $Inn(G)=Aut(G)$, then ...
1
vote
2answers
32 views
GgT, (polynomial) division and finite fields…
Exercise: Let $f,g \in \mathbb{Z}_2[x]$ be the polynomials $f = x^6 + x^5 + x^4 + 1$ and $g = x^5 + x^4 + x^3 + 1$. Has the diophantic equation $f u + g u = x^4 + 1$ solutions $u,v \in \mathbb{Z}[x]$? ...
2
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2answers
48 views
Direct Product of the $G_i $'s
I am a little confused in the interpretation of the product of groups. Here is what's written in my notes:
Given groups $G_1,G_2,...,G_n$, recall that $G_1\times G_2\times ...\times ...
3
votes
4answers
66 views
How to find all elements of $\mathbb{Z}_{4} \times\mathbb{Z}_{4}/\langle(1,1)\rangle$?
I am studying factor groups, and I saw an example that says
Find all the elements of the factor group $\mathbb{Z}_{4} \times \mathbb{Z}_{4}/\langle(1,1)\rangle$.
I know that the order of ...
1
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0answers
21 views
Intersection of prime ideals in Boole ring with identity
Let R be a Boole ring with identity (i.e. $a^2=a$ for all $a$). Then the intersection of all prime ideals in $R$ is known to be equal to $R$'s nilradical, which is $\{0\}$.
Is there a way to show ...
4
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2answers
65 views
Which one of the following groups is decomposable?
A group $(G,+)$ is said to be decomposable if $G$ has two non-trivial subgroups $G_1$ and $G_2$ such that $G=G_1+G_2$ and $G_1 \cap G_2 =$ {$e$}. Then which of the following are decomposable:
(i) ...
12
votes
5answers
696 views
Do we really need polynomials (In contrast to polynomial functions)?
In the following I'm going to call
a polynomial expression an element of a suitable algebraic structure (for example a ring, since it has an addition and a multiplication)
that has the form ...
2
votes
3answers
87 views
Why can't this be a coset?
Let $H$ be a subgroup of $G$ and H is not normal, there are left cosets $aH$ and $bH$ whose product isn't a coset.
My attempt:
$ab H\subset aHbH$ and if H is not normal, if $ah_1bh_2=abh_3$ for all ...
3
votes
1answer
51 views
Automorphism of $A[t]/(t^m)$
Let $A$ be a commutative ring and $t$ an indeterminate over $A$. If $f$ is an automorphism of the ring $A[t]/(t^m)$ satisfying $f(x)\equiv x\pmod{(t)}$ for each $x\in A[t]/(t^m)$ with $m$ a positive ...
3
votes
1answer
41 views
For polynomial prove that $r(x)=0$ or $\deg [r_k(x)]<\deg [b(x)]$
Suppose $a(x),b(x)\in \mathbb R[x]$, $\deg(b(x)) \geq 1$. Show that there exists $m=0,1,\dots$ and $r_0(x),\dots,r_m(x)\in\mathbb R[x]$ such that
$$ a(x)=r_0(x)+r_1(x)b(x) + \cdots + r_m(x)b(x)^m$$
...
1
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1answer
37 views
Consider a Group $G$ of order $20$ such that $G=Aut$($Z_{25},+) \cong (U_{25}, \cdot)$. Analyze the Sylow subgroups in G.
$G=Aut$($Z_{25},+) \cong (U_{25}, \cdot)$
I know that there is one 5-Sylow subgroup and number of $2-Sylow$ subgroups is either $1$ or $5$.
(a) How do I decide whether the number of distinct 2-Sylow ...
2
votes
2answers
63 views
Units on a ring
Let $A$ be a ring such that there exists units $u_1, \dots , u_p : u_1 + \dots + u_p=0$
and $p$ a prime. Did $A$ have a subring isomorphic to $\mathbb{Z}/p \mathbb{Z}$ in general
Thanks in advance.
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2answers
31 views
Question about derivation in Jordan algebra
Let $(G,\circ)$ be a Jordan algebra, then $\sigma:G\to G$ given by $$c\mapsto a\circ(b\circ c)-b\circ(a\circ c),\quad\forall c\in G,$$ is a derivation, where $a$ and $b$ are two fixed elements of $G$.
...
2
votes
2answers
48 views
Rotman's Introduction to to the theory of groups. Exercise 3.45.
Can you give me a hint on the first part of the exercise?
Let $p$ be a prime and let $X$ be a finite $G$-set, where $|G| = p^n$ and $|X|$ is not divisible by $p$. Prove that there exists $x \in X$ ...
1
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2answers
54 views
Questions about projective modules.
Let $P$ be a projective module and $M$ a submodule of $P$. We know that $M$ is also a projective module. Can we conclude that $P=M\oplus N$ for some module $N$? Thank you very much.
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1answer
56 views
Find all ideals of ring $\mathbb{Z}/m\mathbb{Z}$. [closed]
Find all ideals of ring $\mathbb{Z}/m\mathbb{Z}$.
2
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1answer
83 views
What does it mean to “Decide to which group $G$ is isomorphic” for a given group $G$?
I have a homework question which is
Decide to which group $(\mathbb{Z}_n^*,\,\cdot\,)$ is isomorphic (classification of finite abelian groups), for
(i) $n = 9$,
(ii) ...
1
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1answer
26 views
Self Inverse in Integral Domains
I need to show that if R is an integral domain and it has unity then the only elements of R which are inverse to itself are 1 and -1 (with respect to multiplication). But I don't know where to start. ...
1
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3answers
58 views
How to find all ring homomorphisms from $\mathbb{Z}$ to $\mathbb{Z}$
I am trying to find all homomorphisms from $\mathbb{Z}$ to $\mathbb{Z}$. I am looking at a solution, but i do not understand it. Here it is:
Let $\Phi\colon \mathbb{Z}\to\mathbb{Z}$ be a ...
-5
votes
1answer
120 views
Root of a quadratic equation that has modulus $1$
Let us suppose $\alpha \in \mathbb C$ and $|\alpha|=1$ and $\alpha$ satisfies a monic quadratic equation. Then prove that $\alpha^{12} =1$.
Show me the right way to solve this. Thanks in advance.
2
votes
2answers
45 views
Does every element of an integral domain have an inverse?
I am reading a first course in algebra 7th edition written by John B. Fraleigh. I have seen the following two definitions:
1) A field is a commutative ring in which every nonzero element has
...
1
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0answers
38 views
Show that an field extension is algebraic (normal).
Let $A/K$ be a field extension, I wanted to proof:
$A/K$ is normal iff for every irreducible polynomial $P \in K[x]$ which has a root in $A$, the field extension $A$ contains a splitting field for ...
2
votes
1answer
55 views
Example 1.K in A User's Guide to Spectral Sequences
I'm having trouble with Example 1.K, p.25, of John McCleary's book A User's Guide to Spectral Sequences.
Specifically, I don't understand how he defines the "obvious map" in the second ...
1
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1answer
50 views
Viewing groups as objects of the concrete category $\mathsf{Grp}$
Sometimes I ask questions about how structures (groups, topological spaces etc.) ought to be defined, and oftentimes a categorial solution is suggested. Here is a recent example.
Now from my ...
0
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1answer
33 views
Normal matrices with orthogonal basis
we have a theorem that says that each REAL normal matrix can be written in terms of an orthonormal basis, so that it has its eigenvalues down the diagonal and 2x2 matrices of the form $\begin{pmatrix} ...
9
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6answers
252 views
Should every group be a monoid, or should no group be a monoid?
Question: What is more convenient/useful? Writing mathematics as if every group is a monoid, or as if these two classes are disjoint?
Additional discussion. Define a monoid as follows.
Defn 1. A ...
1
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0answers
26 views
The criteria for two abelian extensions to be embedded
Learning class field theory I found this theorem, but I can't prove it or find the solution. I'll be glad to any help.
Let $L$ and $M$ be abelian extensions of $K$. $L \subset M$ if and only if ...
8
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0answers
54 views
Question about the Dedekind completion of a non-archimedean ordered field.
Suppose we have a non-archimedean ordered field $F$. No such field is Dedekind-complete, for that property implies the Archimedean one. But, we can of course fill in the gaps and form the Dedekind ...
3
votes
2answers
62 views
Silliness: $\exists~X~\text{s.t.}~AX=B \iff B\in R(L_A)$
So, I am asked to prove that the system of linear equations $AX=B$ has $\color{black}{a~solution}$ if and only if $B\in R(L_A)$. $R$ denotes the "range of" and $L_A$ is left multiplication by $A$. If ...
7
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1answer
73 views
Let $G$ be a group of order $pq$, with $p$ and $q$ prime. Prove that the order of the center of $G$ is 1 or $pq$.
Let $G$ be a group of order $pq$, with $p$ and $q$ prime. Prove that
the order of the center of $G$ is 1 or $pq$.
Let me start off with what I did:
Assume $G$ is abelian. Then we know ...
3
votes
1answer
26 views
length of composition series
Let $f:M\rightarrow N$ be an injective $R$-module homomorphism. Show that $l(M)\leq l(N)$ where $l(M)$ denote the number of nonzero submodules in a composition series of $M$.
My solution:
Since $f$ ...
5
votes
0answers
91 views
An example of a commutative ring in which every primary ideal is prime
It is clear that every prime ideal in a commutative ring is primary. The converse is false; for example, in the ring $\mathbb{Z}$ the ideal $(p^2)$ is an example of a primary ideal that is not prime ...
1
vote
2answers
48 views
radical of sum of two ideals
$I$ and $J$ are ideals in $k[x_1,\cdots,x_n]$.
Show that $\sqrt{I+J}=\sqrt{\sqrt{I}+\sqrt{J}}$.
I have no idea how to prove it. Can someone help?
2
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1answer
40 views
Space modelled on ring
I am learning some deformation theory. The dimension of the moduli space for some deformation problem is bound by the dimension of (usually) the first cohomology group of the object that is to be ...
1
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1answer
38 views
$P \cap N$ is a Sylow $p$-subgroup of $N$, where $N$ is normal in $G$ and $P$ is a Sylow $p$-subgroup of $G$?
In 'A Course in Group Theory' by Humphreys, Proposition 11.14 says that if $G$ is a finite group, $P$ is a Sylow $p$-subgroup of $G$ and $N$ is a normal subgroup of $G$, then $P \cap N$ is a Sylow ...
2
votes
1answer
20 views
Module isomorphism, simple modules, and quotients
I'm reading R.S. Pierce's Associative Algebras. While proving a preliminary lemma to Nakayema's Lemma, the following is mentioned:
Let $M$, $N$, be two $A$-modules where $N$ is a submodule of $M$ ...
5
votes
3answers
102 views
What is the meaning of the parentheses in $\phi^{-1}\left[\{\phi(g)\}\right]=gH=Hg$?
I am studying homomorphisms is groups and i saw a theorem saying:
For $g$ in a group $G$, the cosets $gH$ and $Hg$ are the same, and collapsed onto the single element $\phi(g)$ by $\phi$. That is, ...
7
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4answers
158 views
Connection between number theory and abstract algebra
I haven't taken abstract algebra yet but I was curious on what connections do number theory and abstract algebra share? Do the proofs of things like Fermat`s little theorem, the law of quadratic ...
0
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2answers
93 views
$GL(n, \mathbb{C})$ is algebraically closed? [closed]
Let $GL(n,\mathbb{C})$ the group of non-singular matrices. Is it algebraically closed? For $GL(1,\mathbb{C})$ is it true; but if I take linear combinations of elements in $GL(n,\mathbb{C})$ with ...
3
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3answers
63 views
Book recommendation for associative algebras
Currently, I am reading David Radford's Hopf Algebra, and I would like to pick up some representation theory of associative algebras as well since my knowledge of them is pretty shallow at the moment.
...
12
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0answers
92 views
A ring isomorphic to its finite polynomial rings but not to its infinite one.
I was messing with the ring $k[x_1,\dots,x_n,\dots]$ of polynomials in numerable many variables in order to solve an exercise of Atiyah, and the following question came to me and made me curious:
...
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2answers
42 views
Proof: let $f:A \to B$ with $f$ bijective, then $f{\restriction_{C} }: C \to B$ is bijective
I need the proof of following:
"let $f:A \to B$ with $f$ bijective, then $f{\restriction_{C} }: C \to B$ is bijective"
Thanks in advance
0
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1answer
46 views
a necessary and sufficient condition that homomorphic image of $R$ is a field
Let $R$ be a commutative ring with unity.Find a necessary and sufficient condition that homomorphic image of $R$ is a field
I am little bit confused about the question. By a necessary and ...
7
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1answer
66 views
Why does the automorphism used to construct the group have to be non-inner?
I have a question on why a particular assumption is made that the automorphism used to construct a certain group be non-inner.
In [Herstein, Topics in Algebra, p. 69], a construction of a nonabelian ...
5
votes
1answer
58 views
How does Tor and the Tensor functor interact?
So I've run into this question while doing some computations and I'm unsure if what I'm trying to show is true. Assume tensors are over $\mathbb{Z}$, is
...
1
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1answer
51 views
Prove that if R and S are nonzero rings then $R\times S$ is never a field.
This question has been asked on here before but I'm looking for some additional insights. The question comes from the section of the Dummit and Foote textbook on the Chinese Remainder Theorem. All of ...
