Tagged Questions
Abstract algebra is the 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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1answer
11 views
Subgroup of roots of unity of a field.
Let $F$ be a field. Show that the set of all $n$th roots of $1$ is a subgroup of $F^\times$.
0
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0answers
6 views
A Confusion regarding semiring
In page number 3 of J.S. Golan's book Semirings and their applications, there is a result which says that
if $R$ is a hemiring and $S$ is a subhemiring of $R$ which is a semiring having ...
0
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0answers
14 views
Well Known Polynomial Division over field
This proerty seems known and intuitive, but for some reason i am not able to manufacture a proof. If $p$ is prime and $d,n$ positive integers, then $(x^{p^d}-x)|(x^{p^n}-x)$ in $(Z/pZ)[x]$ if and only ...
0
votes
1answer
16 views
Give an example of the fact that the lattice of all subgroups need not be modular?
A hint in my book says that one could use D4 and the subgroups M={e,a,a^2,a^3},H={e,a^2} and K={e,b} to show that modularity fails. For reference, modularity means H∨(K∧M)=(H∨K)∧M and it is required ...
0
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1answer
30 views
Commutators Calculus
I was trying to understand the above Corollary but I have a problem, namely why in the second to last line $A_0 \leq \zeta_p(G)$? Any ideas?
Definitions
By recurrence we define $[x,_0\, y]=x$; ...
0
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1answer
78 views
Trascendence of $\displaystyle e^\pi$ and $\displaystyle \pi^e$
Is there some relation between the trascendence of $\displaystyle e^\pi$ and that of $\displaystyle \pi^e$? I mean: the transcendence of one implies the other or the proofs are independent? Thanks.
3
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0answers
34 views
Prove that $G$ has a nontrivial normal subgroup
I wanted to ask if I had done this problem correctly.
Let $G$ be a group of order $pqr$ (for $p > q > r$ primes).
(i) If $G$ fails to have a normal subgroup of order $p$, determine the ...
0
votes
1answer
14 views
tensor product of R-algebra and f.g module [on hold]
$R$ is a commutative noetherian ring. If $S$ is an $R$-algebra, and $M$ a finitely generated $R$-module, is $M\otimes_RS$ finitely generated $S$-module? If this is not true, what can i add to ...
3
votes
1answer
24 views
Embedding of Galois Group
I am trying to prove the following:
Let $E/k$ be a splitting field of $f(x)\in k[x]$ with Galois group $G=\operatorname{Gal}(E/k)$. Prove that if $k^*/k$ is an extension field and $E^*$ is a ...
2
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0answers
34 views
Where can I find an ontology of algebraic structures?
A group is a monoid where every element admits an inverse,
A ring is a monoid under multiplication that distributes over a commutative group
A field is a ring whose non-zero elements form a group ...
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votes
1answer
23 views
$p$ and $q$ are primes. Prove $\forall n,k\in \mathbb N, (p^n\mid q^k⇒p=q)$
I'm having trouble answering this question, can anyone help explain a full solution of this problem? I will be very grateful. Thanks!
0
votes
3answers
19 views
Let n and k be positive integers. If n ≥ (k + 1), then n! + (k + 1) is a composite number
I've been having trouble with finding the proof for this question. Can anybody explain the solution to me? Thanks so much!
1
vote
1answer
38 views
If $X^{p^d}\equiv\text{ mod }f$ for prime numbers $p,d$, then $f\in\mathbb{F}_p[X]$ with $\deg f=d$ is irreducible over $\mathbb{F}_p$
Let $p$ and $d$ be prime numbers and $f\in\mathbb{F}_p[X]$ with $\deg f=d$ and no roots over $\mathbb{F}_p$. I want to show $$X^{p^d}\equiv X\text{ mod }f\;\;\;\Rightarrow\;\;\;f\text{ is irreducible ...
1
vote
2answers
32 views
Question regardles primes and the fundamental theorem of arithmetic
I have been reading through my book of practice proofs and came across this particular question which has stumped me.
p and q are primes. Prove $\forall p \in \mathbb{Z}, \forall k \in \mathbb{Z}, ...
2
votes
0answers
21 views
For vector spaces $V,W$, probe that $\Lambda (V)\otimes\Lambda (W) \cong \Lambda (V\oplus W)$ [duplicate]
For vector spaces $V,W$, probe that $\Lambda (V)\otimes\Lambda (W) \cong \Lambda (V\oplus W)$.
I have tried to use the universal property, but I can not create the necessary linear transformation.
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0answers
38 views
Is set of even integers an integral domain?
Set of even integers forms a commutative ring with no zero divisors.
But it does not have unity.
So is it an integarl domain?
Few books say that integral domain should possess unity and some books do ...
4
votes
2answers
40 views
the nonzero elements of Z3[i] form an abelian group of order 8 under multiplication. Is it isomorphic to Z8??
$\mathbb{Z}/3\mathbb{Z}[i]$ is an integral domain, so its characteristic is a prime number.
But, in order to prove that it is isomorphic to $\mathbb{Z}_8$, we have to show that $\mathbb{Z}_3[i]$ has ...
1
vote
0answers
38 views
Interpreting statements in Lang's Undergraduate Algebra
So, I've been reading this book and I've come across two sentences that I find a little confusing.
On pg. 109:
The polynomial ring $R[t]$ is generated by the variable $t$ over $R$, and $t$ is ...
3
votes
1answer
57 views
Determine Units of a Ring $\mathbb{Z}[\alpha]$
I am trying to determine the units of $Z[\alpha]$ where $\alpha$ satisfies the monic polynomial $\alpha^4+\alpha^3+\alpha^2+\alpha+1$. I found $Z[\alpha] := \lbrace a+b\alpha+c\alpha^2+d\alpha^3\;|\; ...
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1answer
37 views
Finite field has $t^p - 1 = (t-1)^p$.
Consider the field $\mathbb{Z} / p \mathbb{Z}$, where $p$ is a prime number.
I'm reading Lang's Undergraduate Algebra, and he asserts that $t^p - 1 = (t-1)^p$. Why is this true?
Thanks!
3
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0answers
24 views
Determining if a given equation is solvable given a set of ultra-radicals
So suppose someone is armed with the tools of standard arithmetic, exponents (and of course that comes along with roots)
AS WELL AS a set of inverses for some polynomials which are not solvable using ...
0
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1answer
38 views
Allow $2 \Bbb N$ to denote the even integers $> 0$.
Please help!
Allow $2\Bbb N$ to denote the even integers $> 0$. Say $a \in 2\Bbb N$ is irreducible if there are no numbers $b, c \in 2\Bbb N$ so that $a = bc$.
(1) Show that if $n$ is an odd ...
1
vote
1answer
37 views
Double cosets and conjugation
Let G be a group and $h,g \in G$ with $SgT=ShT$
Show that the subgroups $gTg^{-1}\cap S$ and $hTh^{-1}\cap S$ are conjugated in S.
Two subgroups $U,V$ are conjugated if $\phi(U)=gUg^{-1}=V$, right?
...
7
votes
3answers
752 views
Meaning of math symbol ~
Segment of Example:
t = ...
More usefully, we have:
t ~ n*log(n)
Note: ~ means "similarity" like in geometry, same shape but not same size. How is it interpreted here?
Edit: yes, t depends on n
...
1
vote
0answers
16 views
Graduations and filtrations for localizations
I'm trying to answer the following questions:
Let $A$ be a (not necessarily commutative) $\mathbb{Z}$-graded ring and $S$ a multiplicative subset of $A$ such that $AS^{-1}$ exists. Is $AS^{-1}$ a ...
1
vote
1answer
37 views
exact sequence problem
Let $B_1 \stackrel{f}{\to} B \stackrel{g}{\to} B_2 \to 0$ be an exact sequence. For any module $M$ the sequence $$0 \to \mbox{Hom}(B_2,M) \stackrel{g*}{\to} \mbox{Hom}(B,M) \stackrel{f*}{\to} ...
0
votes
0answers
29 views
Let $K$ a field, $\operatorname{char}(K)=0$. Let $V$ a vector space over $K$, $\dim(V) \geq 1$, and be $f$ a $n$-tensor. Prove that $f \wedge f =0$
Let $K$ a field, with $\operatorname{char}(K)=0$. Let $V$ a vector space over $K$, $\dim(V) \geq 1$, and be $f$ a $n$-tensor ($f \in {\mathcal T}_n(V):=\Lambda^{n}(V)$), i.e., $f$ is an multilinear ...
-1
votes
0answers
13 views
basis vectors of a 2D lattice plane in a 3D lattice
I know the basis vectors of the three-dimensional lattice $\Lambda = \{\mathbf{b_1}, \mathbf{b_2}, \mathbf{b_3} \}$. I also know the equation of the plane in this 3D lattice, suppose $Ax + By + Cz = ...
1
vote
2answers
52 views
Complex analysis - existence of field $\mathbb{C}$
In the following: $\mathbb{F}$ is defined to be a field containing $\mathbb{R}$ and in which the equation $x^{2}+1=0$ can be solved.
Then define a set $\mathbb{C}$ to be subset of $\mathbb{F}$ whose ...
1
vote
1answer
26 views
Structure Theorem for Finite Commutative Rings with unity [duplicate]
The Structure Theorem for Finite Commutative Rings with unity
state that:
A finite commutative ring $R$ with multiplication identity
is isomorphic to a direct sum of local rings.
Suppose all the ...
0
votes
1answer
48 views
Prove $Z_{p}$ are prime fields,where $p$ is prime numbers
show that $Z_{p}$ are prime fields,where $p$ is prime numbers.
maybe this problem is old,But I look for some book,and can't find it,someone know which book have this problem proof?
because I know ...
0
votes
1answer
24 views
Proving closure of unit space of a Hausdorff groupoid
For Hausdorff topological groups, the set $\{e\}$ containing only the identity is closed. This is because Hausdorff implies T1 which implies singletons are closed.
For topological groupoids, defined ...
1
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0answers
44 views
Name of an aglebraic structures $(A,*,\cdot)$ weaker than semirings.
I have a set $A$ with two binary operations on it
$(A,*,\cdot)$
STRUCTURE A
$(A,*)$ is not commutative, is not associative, it has not an identity
$(A,\cdot)$ is a commutative group
$(a*b)\cdot ...
2
votes
0answers
28 views
Angles between adjacent roots in a reduced root system.
Let $R$ be a reduced root system. ($R$ is a finite set spanning $V$, $\alpha \in R \rightarrow -k\alpha \in R$ iff $k=1$, $s_{\alpha}(R)=R$, $s_{\alpha}(\beta)-\beta=k\alpha$ whit $k$ integer).
...
7
votes
3answers
611 views
Masters' thesis in group theory
I would like some ideas on topics in group theory which would be suitable for a masters' thesis. What sort of problems would be suitable for this level? Because it is at masters' level, no original ...
0
votes
0answers
25 views
Type of isomorphism
Let K algebraic closed field, M a simple A-module where A a K-algebra then $ End(M) \cong K $.
My question is what kind of isomorphism there is between the two objects.There is a correspondence for $ ...
1
vote
1answer
30 views
Which is a subring? Which is an ideal?
We are having ring $\mathbb{Z}[\sqrt{-6}]$.
Which of the sets is subrings of $\mathbb{Z}[\sqrt{-6}]$ and which are ideals?
$\mathbb{Z}+5\mathbb{Z}[\sqrt{-6}]$
$5\mathbb{Z}+\mathbb{Z}[\sqrt{-6}]$
...
8
votes
2answers
115 views
Homogeneous groups
Let's call a group $G$ homogeneous if for every two distinct, non-identity elements $a$ and $b$ there is an automorphism $\phi$ of $G$ such that $\phi(a)=b$.
Examining this definition, we can see ...
0
votes
0answers
28 views
Is there a reason behind the order of operations? [duplicate]
I am curious as to why there is a certain defined "order" or "hierarchy" for the common operations (e.g. addition, subtraction, multiplication, etc). Specifically, I am referring to PEMDAS ...
0
votes
1answer
31 views
What is the difference between a bijection and a reversible transformation?
I was reading http://arxiv.org/abs/quant-ph/0101012v4 and one of the axioms is that there needs to be a continuous reversible transformation between states. What is the difference between that and a ...
2
votes
4answers
112 views
If $a+\sqrt{b}$ is a root of a polynomial equation with integer coefficients, so is $a-\sqrt{b}$
I tried to use the Briot-Ruffini method but it didn't work.
The question I need help is:
"Prove that, if a polynomial equation with integer coefficients has the irrational number $a+\sqrt{b}$ as a ...
3
votes
1answer
42 views
Maximal linearly independent sets in a f.g. module
Suppose $M$ is a finitely generated module over a commutative unital ring $R$.
Is it true that every maximal linearly independent set in $M$ has the same size?
What is the most general condition ...
0
votes
1answer
28 views
Finite Cyclic group isomorphic to C
If $G$ is a cyclic group and $|G|=n$ ($n\in \Bbb Z_{\ge 1}$), then $G\cong C_n$.
Why does this statement not hold in both directions, i.e an if and only if statement?
1
vote
1answer
36 views
hom and exact sequence
Let $$ 0 \longrightarrow \operatorname{Hom}(M,Β_1) \stackrel{f^*}\longrightarrow \operatorname{Hom}(M,Β) \stackrel{g^*}\longrightarrow \operatorname{Hom}(M,Β_2) $$ be an exact sequence for any ...
1
vote
0answers
39 views
Automorphim of $\mathbb C$ that exchanges two transcendental elements.
Consider the following proposition:
Let $\alpha$ and $\beta$ be two transcendental elements over $\mathbb
Q$, then there exists an automorphism
$\sigma\in\textrm{Aut}(\mathbb C)$ such that ...
2
votes
1answer
41 views
Prove that $g:R/\ker(φ)\to \operatorname{Ιm}(\varphi)$ where $ g(\ker \varphi+r)=\varphi(r)$ is $1$-$1$
Given a ring homomorphism $\varphi:R\to S$, how to prove that:
$g:R/\ker(φ)\to \operatorname{Ιm}(\varphi)$ where $ g(\ker \varphi+r)=\varphi(r)$ is $1$-$1$
I know that function is an isomorphism ...
0
votes
1answer
41 views
Characterization of the kernel and cokernel of the natural homomorphism between a module and its double dual. [on hold]
Let $R$ be a Noetherian ring and $M$ a finite $R$-module. Suppose $$ G \overset{\varphi}{\rightarrow} F \to M \to 0$$ is exact where $F,G$ are finite free modules. Suppose ...
0
votes
1answer
41 views
Isometries Proof
In my attempt to prove this result I got up to the point before the part I have highlighted in orange. Is the part highlighted in orange actually necessary?
1
vote
0answers
27 views
Is there anything behind the “$m + N$” notation for elements in the factor structure $M/N$?
Or to be more precise,
Is there anything behind the “$m + N$” notation for elements in the factor structure $M/N$ other than a hint at the set-theoretical realization?,
for example, if $M$ and ...
2
votes
0answers
28 views
Factoring over ring of power series [duplicate]
How would we factor $6+x$ over $\mathbb{Z}[[x]]$, the ring of formal power series with integer coefficients? Proving things are prime is easy, but factoring a nonprime is difficult. Thanks in advance.
...
