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.
0
votes
5answers
58 views
Prove that if $G$ is abelian, then $H = \{a \in G \mid a^2 = e\}$ is subgroup of $G$ [duplicate]
Let $G$ be an abelian group. Prove that $H = \{a \in G \mid a^2 = e\}$ is subgroup of $G$, where
$e$ is the neutral element of $G$.
I need some help to approach this question.
4
votes
4answers
65 views
Irreducible Polynomial in $\mathbb F_{256}$.
Let $\mathbb F_{256}$ be the finite field with $2^8 = 256$ elements. Consider the polynomial over this field
$$
x^2 + x + 1.
$$
I wanted to know if it is irreducible, so I calculated it for all ...
2
votes
0answers
38 views
Structure Theorem for finitely generated modules over PIDs [duplicate]
Let $A$ be a real 4 by 4 matrix. Supose $i,-i$ are the eigenvalues of $A$. Show that there exists an invertible matrix $P$ such that $PAP^{-1}$ is either
$$\begin{pmatrix}
0&-1&0&0\\
...
0
votes
2answers
57 views
Find Aut$(G)$, Inn$(G)$ and $\dfrac{\text{Aut}(G)}{\text{Inn}(G)}$ for $G = \mathbb{Z}_2 \times \mathbb{Z}_2$
Find Aut$(G)$, Inn$(G)$ and $\dfrac{\text{Aut}(G)}{\text{Inn}(G)}$ for $G = \mathbb{Z}_2 \times \mathbb{Z}_2$.
Here is what I have here:
Aut$(G)$ consists of 6 bijective functions, which maps $G$ to ...
4
votes
3answers
82 views
Let $G$ be a finite group with $|G|>2$. Prove that Aut($G$) contains at least two elements.
Let $G$ be a finite group with $|G|>2$. Prove that Aut($G$) contains at
least two elements.
We know that Aut($G$) contains the identity function $f: G \to G: x \mapsto x$.
If $G$ is ...
3
votes
1answer
30 views
When is the quotient algebra of a unital C* algebra helpful?
Let $\mathcal A$ be a unital C* algebra.
Which properties does $\mathcal B \subset \mathcal A$ has to have for it to make sense to form the quotient algebra $\mathcal A / \mathcal B$?
In cases ...
1
vote
1answer
25 views
how to show that $f(x)$ can be expressed uniquely as follows: $f(x)=\prod_{i=1}^k[f_i(x)]^{n_i}$
Let $f(x)\in F[x],~F$ being a field, be monic. Then how to show that $f(x)$ can be expressed uniquely as follows:
$$f(x)=\prod_{i=1}^k[f_i(x)]^{n_i}$$
for some monic irreducible polynomial ...
2
votes
1answer
53 views
Two problems about Structure Theorem for finitely generated modules over PIDs
1) Let $A$ be a real 4 by 4 matrix. Supose $i,-i$ are the eigenvalues of $A$. Show that there exists an invertible matrix $P$ such that $PAP^{-1}$ is either
$$\begin{pmatrix}
0&-1&0&0\\
...
6
votes
4answers
61 views
Any commutative associative operation can be extended to a function on nonempty finite sets
This is a fact we use very frequently in general mathematics when we write such notations as $1+2+3+4$: since we know that $+$ is commutative and associative, we can just "drop the parentheses" and ...
2
votes
0answers
31 views
Find the factorization of the polynomial as a product of irreducible [duplicate]
Find the factorization of the polynomial $x^5-x^4+8x^3-8x^2+16x-16$ as a product of irreducible on rings $R[x]$ and $C[x]$
Testing with the simplest possible root in this case, $P(1)=0$
Applying the ...
1
vote
1answer
22 views
Algebra / Equation with 2 or more vaiables
A train moves past a telegraph post and a bridge 264 m long in 8 seconds and 20 seconds
respectively. What is the speed of the train?
I'm really confused if there are 2 or more variables. Can ...
3
votes
1answer
42 views
Why is $[\mathbb{Q}(e^{2\pi i/p}):\mathbb{Q}] = p-1 $?
I know that $e^{2\pi i/p}$ is a root of $x^{p}-1$ and we can write:
$ x^{p}-1=\left(x-1\right)\left(\sum_{i=0}^{p-1}x^{i}\right) $
So $e^{2\pi i/p}$ is a root of $\sum_{i=0}^{p-1}x^{i}$, which means ...
5
votes
1answer
50 views
Is the set of submonoids of $(\Bbb N,+)$ countable?
Having the monoid $(\Bbb N,+)$, I wonder if there are countable many submonoids. There are obviously infinitely many since $S_n = \{kn \mid k \in \Bbb N\}$ is a submonoid for any $n \in \Bbb N$.
My ...
1
vote
1answer
36 views
Find the factorization of the polynomial as a product of irreducible on rings R[x] and C[x]
Find the factorization of the polynomial $x^5-x^4+8x^3-8x^2+16x-16$ as a product of irreducible on rings $\Bbb R[x]$ and $\Bbb C[x]$
Testing with the simplest possible root in this case, $P(1) = 0$
...
2
votes
0answers
44 views
Irreducible polynomial roots and representations for Galois field elements in normal basis
I am trying to understand some topics in the literature and came across the following problem. Say I have a field $GF(2^4)$ defined by the irreducible polynomial $r(z) = z^4 + z + 1$, and I want to ...
0
votes
1answer
113 views
Is $\{a + bi: a, b \in \mathbb{F}_3\}$ a field?
Is $P = \{a + bi: a, b \in \mathbb{F}_3\}$ a field? This is the question I am having.
First, I list the elements in $\mathbb{F}_3$, which consists of $\overline{0}$, $\overline{1}$ and ...
3
votes
2answers
35 views
How to deal with polynomial quotient rings
The question is quite general and looks to explore the properties of quotient rings of the form $$\mathbb{Z}_{m}[X] / (f(x)) \quad \text{and} \quad \mathbb{R}[X]/(f(x))$$
where $m \in \mathbb{N}$
...
6
votes
1answer
51 views
Calculating in quotient ring of $\mathbb{R}[X]$
Part of an old Oxford exam (1992 A1)
We want to find which elements of the quotient ring $\mathbb{R}[X]/(x^3-x^2+x-1)$ are equal to their own square.
Now, we note first that ...
5
votes
1answer
34 views
Generalising the Chinese Remainder Theorem
We have that for $I,J$ ideals of some ring $R$ with $R=I+J$, $$\frac{R}{I\cap J} \cong \frac{R}{I} \times \frac{R}{J}$$
My question is whether the analogous expression for three ideals $I,J,K$ where ...
5
votes
1answer
39 views
Express $4+\sqrt{-2}$ as a product of irreducibles
This is part of an old Oxford Part A exam paper. (1992 A1)
Suppose we equip $R=\mathbb{Z}[\sqrt{-2}]$ with the Euclidean function $d$ defined by $$d(m+n\sqrt{-2})=|m+n\sqrt{-2}|^2$$
I want to ...
-6
votes
0answers
40 views
Direct products and Semidirect products [closed]
If $G$ is a group of order $35$
a) Explain why $G$ is a direct product $\Bbb Z_7 \times\Bbb Z_5$, or is one of the possible semidirect products $\Bbb Z_7 \times \Bbb Z_5$.
b) Determine all possible ...
-4
votes
0answers
39 views
Group automorphisms and Sylow subgroups [closed]
Consider the automorphism group G = Aut(Z25,+) isomorphic to (U25,*)
a) What is the order of G? Is G cyclic? What are the orders of its Sylow subgroups?
b) What are the isomorphism types of the ...
-5
votes
0answers
36 views
Group direct products [closed]
The direct product $\Bbb Z_{45}\times\Bbb Z_{98}$ is cyclic and isomorphic to $\Bbb Z_{4410}$ because $\gcd(45,98) = 1$; furthermore the element $1 = \left([1]_{45},[1]_{98}\right)$ is a cyclic ...
-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
71 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
56 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 ...
11
votes
1answer
73 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
91 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
35 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
74 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
32 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
45 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
$$\require{AMScd}
\begin{CD}
1 @>>> 2\\
@V{}VV @V{}VV \\
3 @>>> 4 @>>> 5
...
4
votes
1answer
55 views
Another basic short exact sequence problem
In the following commutative diagram of R-modules, all of the rows and columns are exact. Prove that $K$ is isomorphic to $L$.
\begin{array}{ccccccccccc} &&&&&&&&0 ...
3
votes
3answers
73 views
very basic short exact sequence problem
Given a short exact sequence $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow0$ and $f:A \rightarrow B, g: B \rightarrow C$, why is $C$ isomorphic to $B/A$? All I can show is that $C$ is ...
5
votes
0answers
49 views
Why are the p-adic integers a linearly ordered group? [duplicate]
In a previous question, someone suggested the p-adic integers as an example of a non-archimedean linearly ordered group.
I'm not sure why these are linearly ordered - specifically, it doesn't seem to ...
4
votes
0answers
48 views
Free algebra over $\mathbb{Z}/N\mathbb{Z}$
Let $A$ be a commutative finite free $\mathbb{Z}/N\mathbb{Z}$ algebra of rank 2 with unit discriminant.
I have two questions :
1) Why is it true that $A/pA$ is isomorphic to either ...
1
vote
1answer
45 views
Column entries of a matrix sum to zero, so what are the properties?
What kind of properties does a matrix whose column entries sum to zero have?
$$ \begin{pmatrix} a_{11} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots \\
a_{m1} & \cdots & ...
6
votes
5answers
114 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
27 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
40 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
29 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
42 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
86 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 ...
4
votes
1answer
63 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
94 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
42 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
32 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
58 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 ...
