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.
-4
votes
0answers
45 views
Dihedral groups [closed]
Determine the center $Z(G)$ for the dihedral group $G = D_n$ for $n$ greater or equal to $3$. The answer will depend on whether $n$ is even or odd.
Please be precise, thanks in advance
-2
votes
0answers
45 views
Abstract Algebra automorphisms and isomorphisms [closed]
Consider the automorphism group $\mathrm{Aut}(Z_{16},+)$ isomorphic to $(U_{16},\times)$.
a) By examining the cyclic subgroups in $U_{16}$ show that $\mathrm{Aut}(Z_{16},+)$ is isomorphic to ...
5
votes
1answer
66 views
Order of elements in a group.
Corollary 4.6.8. There is a group $G$ of order $n^3$ given by $G = \{b^ic^ja^k | 0 ≤ i, j, k < n\}$, where $a$, $b$, and $c$ all have order $n$, and $b$ commutes with $c$, $a$ commutes with $c$, ...
1
vote
4answers
55 views
Subrings of $\mathbb{Q}$
Let $p$ be prime. Suppose $R$ is the set of all rational numbers of the form $\frac{m}{n}$ where $m,n$ are integers and $p$ does not divide $n$.
Clearly then $R$ is a subring of $\mathbb{Q}$.
I now ...
10
votes
1answer
64 views
Uniformly solvable families of polynomials
It is a famous theorem that there is no "quintic formula", i.e. there is no formula which expresses the roots of a quintic polynomial $x^5+a_4x^4+\cdots+a_0$ in terms of $a_4,\ldots,a_0$ and rational ...
6
votes
2answers
87 views
Some Results in $\mathbb{Z} [\sqrt{10}]$
This is a question from an old Oxford undergrad paper on calculations in $\mathbb{Z} [\sqrt{10}]$. We equip this ring with the Eucliden function $d(a+b\sqrt{10})=|a^2-10b^2|$. I want to prove the ...
2
votes
1answer
34 views
Minimal Polynomials Annihilating an Abelian Torsion-Free Group
Let $A$ be an abelian torsion-free group. Let $\theta \in\operatorname{Aut}A$. Assume that $\theta$ has a finite period in $\operatorname{Aut} A$, say $n$. Obviously $\theta^n-1$ annihilates $A$ (i.e. ...
3
votes
1answer
57 views
Non-commutative rings without identity
I'm looking for examples (if there are such) of non-commutative rings without multiplicative identity which have the following properties:
1) finite with zero divisors
2) infinite with zero divisors
...
2
votes
1answer
28 views
Question about the definition of representability of a quadratic form
Say I have an integral quadratic form $q$ on $\mathbb{Z}^r$ and another integral binary quadratic form $\tilde{q}$. What does it mean for $q$ to primitively represent $\tilde{q}$? I can't seem to find ...
7
votes
1answer
38 views
Specific projective dimension of a module over bound quiver
Suppose $K$ is an algebraically closed field, and $A$ is the algebra presented by the quiver
$$\require{AMScd}
\begin{CD}
1 @>>> 2\\
@V{}VV @V{}VV \\
3 @>>> 4 @>>> 5
...
4
votes
1answer
53 views
Another basic short exact sequence problem
In the following commutative diagram of R-modules, all of the rows and columns are exact. Prove that $K$ is isomorphic to $L$.
\begin{array}{ccccccccccc} &&&&&&&&0 ...
3
votes
3answers
70 views
very basic short exact sequence problem
Given a short exact sequence $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow0$ and $f:A \rightarrow B, g: B \rightarrow C$, why is $C$ isomorphic to $B/A$? All I can show is that $C$ is ...
5
votes
0answers
46 views
Why are the p-adic integers a linearly ordered group? [duplicate]
In a previous question, someone suggested the p-adic integers as an example of a non-archimedean linearly ordered group.
I'm not sure why these are linearly ordered - specifically, it doesn't seem to ...
4
votes
0answers
47 views
Free algebra over $\mathbb{Z}/N\mathbb{Z}$
Let $A$ be a commutative finite free $\mathbb{Z}/N\mathbb{Z}$ algebra of rank 2 with unit discriminant.
I have two questions :
1) Why is it true that $A/pA$ is isomorphic to either ...
0
votes
1answer
24 views
Column entries of a matrix sum to zero, so what are the properties?
What kind of properties does a matrix whose column entries sum to zero have?
$$ \begin{pmatrix} a_{11} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots \\
a_{m1} & \cdots & ...
6
votes
5answers
112 views
Strong characterization of $\mathbb C$ with respect to $\mathbb R$
$\mathbb R^2$ is not a field, but $2$-tuple arithmetic rules like $(a,b)(c,d)=(ac-bd,ad+bc)$ coupled with $\mathbb R^2$ make it a field, but are there other rules than $(a,b)(c,d)=(ac-db,ad+bc)$ ...
0
votes
1answer
26 views
Sufficient condition for reducibility of polynomial $f(x,y)$
[Dual to
this question]
Suppose we have a non-constant polynomial $f(x,y)\in\mathbb{Q}[x,y]$ such that the following two conditions are satisfied:
1) For every $x_0\in\mathbb{Q}$, the polynomial ...
2
votes
1answer
39 views
Sufficient condition for irreducibility of polynomial $f(x,y)$
Suppose we have a non-constant polynomial $f(x,y)\in\mathbb{Q}[x,y]$ such that the following two conditions are satisfied:
1) For every $x_0\in\mathbb{Q}$, the polynomial $f(x_0, y)\in\mathbb{Q}[y]$ ...
0
votes
3answers
26 views
$n$-linear alternating form with $\dim{V}<n$ $\overset{?}{\text{is}}$ the $0$-form
Prove that every $n$-linear alternating form on a vector space of dimension
less than $n$ is the zero form.
0
votes
0answers
38 views
Linear Combinations of Irrational Numbers: An Analysis on Architecture
Under what condition(s) is
$$ k_1\omega_1+\cdots + k_n\omega_n=c,$$
where $k_i\in\mathbb{R\setminus Q}$ and $\omega_i, c\in \mathbb{Q}$?
I'm essentially trying to show that this is the case only so ...
5
votes
3answers
79 views
Quotient rings of Gaussian integers $\mathbb{Z}[i]/(2)$, $\mathbb{Z}[i]/(3)$, $\mathbb{Z}[i]/(5)$,
I am studying for an algebra qualifying exam and came across the following problem.
Let $R$ be the ring of Gaussian Integers. Of the three quotient rings
$$R/(2),\quad R/(3),\quad R/(5),$$
one ...
2
votes
0answers
38 views
Residue fields of gaussian primes
I just started with algebraic number theory and need some help: In the ring of Gaussian integers a prime $p$ with $p=1\bmod 4$ splits into a product of two irreducible elements $(a+bi)(a-bi)$, with ...
4
votes
2answers
91 views
Finite abelian $2$-group
If $G$ is a finite abelian group such that $o(x)=2$ for all $x \neq e$ and $|G|=2^n$ for some $n\in\mathbb N$, prove that $G \cong \mathbb{Z}_2\times\cdots\times\mathbb{Z}_2$ ($n$ factors).
Any ...
1
vote
1answer
39 views
$n$-linear form: An Interpretation
What is a good example of an $n$-linear form that is more familiar to a student learning at an elementary level?
EDIT:
I'm just trying to show that every $n$-linear alternating form on a vector ...
2
votes
2answers
31 views
Set of Homomorphisms as an $R-$ module
$\require{AMScd}$
I'm reading A first course of homogocial algebra by D.G. Northcott, and I don't quite get the Example 1 on page 25. Here's what it says:
Example 1
Let the module $A$ belongs to ...
0
votes
2answers
50 views
Help in a proof of a result in Hungerford's book
I need help to proof the last part of this corollary:
I didn't understand the part (IV) because the author proves just the canonical projection case and the statement says "every nonzero ...
4
votes
1answer
55 views
How to compute Nakayama functor explicitly?
I am reading the book Elements of representation theory of associative algebras, volume 1. I have some questions related to the Nakayama functor. On page 113-114 of the book, the Auslander-Reiten ...
15
votes
1answer
160 views
Non-associative version of a group satisfying these identities: $(xy)y^{-1}=y^{-1}(yx)=x$
The following identities are a consequence of the group axioms.
$$(xy)y^{-1}=x,\quad y^{-1}(yx)=x$$
Notice we haven't mentioned an identity element, and that the above identities make sense even in ...
6
votes
2answers
72 views
Zeros of a cubic polynomial with rational coefficients
While discussing a related problem, one of my friends came out with a question as follows:
Is it possible that a cubic polynomial $p(x) \in \Bbb{Q}[x]$ has all of its zeros to be both real and ...
1
vote
1answer
34 views
Followup to question concerning $N_G(H) / C_G(H) \cong B \leq \mathrm{Aut}(H)$
Prove $H \leq G$ implies $N_G(H) / C_G(H) \cong B \leq \mathrm{Aut}(H)$ where $B$ is some subgroup of $\mathrm{Aut}(H).$
I did see this question:
For $H \leq G$, showing that $N_G(H)/C_G(H) \leq ...
0
votes
1answer
43 views
Do there exist any polynomial rings with nonzero prime ideals that are not maximal?
I know that with $F$ a field, $F[x]/(f(x))$ is a field iff $f(x)$ is irreducible in $F$. Due to the fact that in a UFD irreducible elements are necessarily prime, we would have that $(f(x))$ is both ...
4
votes
0answers
34 views
Lattices as invertible module
Let $E$ be an etale algebra over $\mathbb{Q}$. In other words, $E$ is a finite sum of number fields. Let $L$ be a lattice in $E$, and $R$ the order associated to $L$. More explicitly, $$R=\{ e\in ...
2
votes
0answers
35 views
Field extension
I need to prove that if $F$ is a field and $u=\frac{f(t)}{g(t)} \in F(t)$ (where $f,g$ are coprime in $F[t]$) then $[F(t):F(u)]=\max(\deg f,\deg g)$.
I know I have to prove that $ug(x)-f(x)$ is ...
8
votes
2answers
34 views
The index of $\xi_4^*$ in $\xi_4$
Just seeing if i'm right:
With the set of solutions for $z^4=1$: $\xi_4=\{1,i,-1,-i\}$, one can construct the group of the $4$th roots of unity: $(\xi_4,\cdot_\mathbb{C})$ and its multiplicative ...
3
votes
1answer
27 views
Show that the subgroup $s(K) \subset H \rtimes_{\alpha} K$ is normal if and only if $\alpha: K \rightarrow Aut(H)$ is the trivial homomorphism.
Show that the subgroup $s(K) \subset H \rtimes_{\alpha} K$ is normal if and only if $\alpha: K \rightarrow Aut(H)$ is the trivial homomorphism, where $s : K \rightarrow H \rtimes_{\alpha} K$ is given ...
2
votes
1answer
73 views
How can I show that $x^4+6$ is reducible over $\mathbb{R}$?
How can I show that the polynomial $x^4 + 6$ is reducible over $\mathbb{R}$ without explicitly finding factors?
I was trying to find a non-prime ideal that would generate it but I'm kind of lost as ...
0
votes
2answers
52 views
Rings | Homomorphisms | Units
Question
Show that if $f :R\rightarrow S$ is a homomorphism, and if $a$ is a unit of $R$, then $f(a)$ is
a unit of $S$. Show, in fact, that $f(a^{−1}) = f(a)^{−1}$ for any unit $a$ of $R$.
Attempt
...
1
vote
0answers
25 views
A Nonzero Alternating Bilinear Form on the Space $P_1(F)$ Over $F$
Can anybody think of an example of a nonzero alternating bilinear form on the space $P_1(F)$ over $F$.
$F$ is a general field like $\mathbb{R}$ or $\mathbb{C}$.
$P_1(F)$ is the set of all ...
0
votes
1answer
35 views
Bilinear Forms: An Initial Condition Proof
Let $B$ be a bilinear form on a finite dimensional vector space $V$. Suppose that for any nonzero vector $v \in V$ there exists a $w \in V$ such that $B(v, w)\neq 0$. Prove that for any linear ...
2
votes
0answers
23 views
The normalizer of a proper sub-algebra properly contains the sub-algebra in a nilpotent lie algebra.
So I am just making my way in to the theory of Lie Algebras. The question at hand comes from page 14 of Humphreys' "Introduction to Lie Algebra and Representation Theory"
Given a finite dimensional ...
1
vote
1answer
54 views
filtration on the (co)homology of a space from the filtration of a space
Fix $n\!\in\!\mathbb{N}$. Let $X$ be a topological space and $X_0\subseteq X_1\subseteq X_2\subseteq \ldots$ subspaces of $X$. Let $\iota_k:X_k\rightarrow X$ be the inclusion. Let
...
2
votes
2answers
86 views
Under what conditions is a zero divisor element $a$ in commutative ring $R$ nilpotent?
Suppose that $R$ is a commutative ring with identity $1$
Let $a\in R$ with $ab=0$ for some $b\ne0$.
Under what conditions $a$ must be also nilpotent?
6
votes
1answer
44 views
Given $G$, when can we find a division ring $R$ with $R^*=G$?
This is motivated by a characterization of finite cyclic groups, in which one proves
Let $G$ be a finite group. If $\#\{g\in G\colon g^d=e\}$ is at most $d$, then $G$ is cyclic.
The proof is ...
2
votes
2answers
57 views
question on subgroups of prime order
Let $G$ be a group and let $\,H,\, K\,$ be subgroups of $\,G,\,$ each of order $\,p,\,$ where $\,p\,$ is prime.
Show that either $\,H\cap K =\{e\},\,$ or $\,H=K.\,$
Is the result true if ...
3
votes
2answers
36 views
Covariant functor, and left exact
I'm reading A first course of Homological Algebra by Northcott, and there is something that the author said it was straightforward. But for some reason, I just don't see the straightforwardness of it.
...
3
votes
2answers
53 views
complete the table by providing an example of a binary operation $*$ defined on $\{a , b ,c\}$
I have a problem with one of my questions. The question is:
complete the table by providing an example of a binary operation $*$ defined on $\{a , b ,c\}$
such that $*$ is commutative and has the ...
0
votes
0answers
29 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 $F[x_1,...,x_n]$, where $F$ is a field , is exactly $n$. And by Koszul Complex, I know its global dimension is greater than or ...
3
votes
5answers
82 views
Guides/tutorials to learn abstract algebra?
I recently read up a bit on symmetry groups and was interested by how they apply to even the Rubik's cube. I'm also intrigued by how group theory helps prove that "polynomials of degree $\gt4$ are not ...
2
votes
2answers
59 views
Some question on localization of polynomial ring
Let $S=A[x_1,\dots, x_r](r \geq 2)$ be a polynomial ring where $A$ is a commutative ring.
Then is it true that $S=\bigcap_{i=1}^r S_{x_i}$? If $S$ is $A$-algebra and $x_i$ are not zero divisors, then ...
2
votes
0answers
22 views
Questions about coalgebras
I have a few questions about coalgebras:
Suppose $C$ and $D$ are coalgebras over the field $k$, and let $f:C\to D$ be a coalgebra map.
(1) If $V$ is a subcoalgebra of $D$, does it follow that ...
