Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, among other topics.

learn more… | top users | synonyms (1)

1
vote
1answer
38 views

A polynomial's irreducibility in $\Bbb{Z}_p$

Show that if $f$ is irreducible in $\Bbb{Z}_p[x]$ then $f$ divides $x^{p^n} - x$ for some $n \in N$. I know that: $f$ is irreducible, so $F = \Bbb{Z}_p / {\left\langle f\right\rangle}$ is a ...
3
votes
1answer
48 views

Stalks of the sheaf of total quotient rings

Let $X$ be a scheme, for each $U$ open in $X$, let $S(U)$ be the set consisting of elements of $O_X(U)$ whose image in $O_{X,p}$ is a non-zerodivisor for every $p\in U$. In particular, if $U = ...
0
votes
2answers
32 views

Proving that cosets partition a group

I have read from multiple sources the statement that the two statements are equivalent: If H is a subgroup of G then the left cosets of H in G partition G Let $g_1H, g_2H$ be two cosets of $H$ in ...
0
votes
1answer
79 views

Proof of The Basis Theorem in Linear Algebra [closed]

So I saw the following proof of the Basis Theorem in Leon Simon's "An Introduction to Multivariable Mathematics". I was wondering if anyone could help explain what is happening in it. I understand ...
1
vote
3answers
36 views

Describe Cosets of subgroup $H= \lbrace 2^n : n \in \mathbb{Z} \rbrace$ of $\mathbb{R}^*$

"A Book of Abstract Algebra" presents this exercise: Describe the cosets of the sub-group: Subgroup $H= \lbrace 2^n : n \in \mathbb{Z} \rbrace$ of $\mathbb{R}^*$. EDIT #2 Thanks to Matt ...
3
votes
2answers
2k views

Ideal generated by 3 and $1+\sqrt{-5}$ is not a principal ideal in the ring $\Bbb Z[\sqrt{-5}]$

Show that the ideal generated by 3 and $1+\sqrt{-5}$ is not a principal ideal in the ring $\Bbb Z[\sqrt{-5}]$. I fail to understand how can 3 and $1+\sqrt{-5}$ generate an ideal. ...
1
vote
1answer
26 views

Quadratic Extensions over Field of Characteristic $\neq 2$.

The following is an example taken from Dummit and Foote. Let $F$ be a field of characteristic $\neq 2$ and let $K$ be an extension of $F$ of degree 2, $[ K : F]=2$. Let $\alpha \in K$ not ...
1
vote
0answers
88 views

Smallest Two-Sided Nearring

For those who are unfamilar with nearrings, here is a definition. A two-sided nearring is a triplet $(N,+,\cdot)$, where $(N,+)$ is a group and $(N,\cdot)$ is a semigroup such that we have both left ...
2
votes
0answers
113 views

Left Ideals and Two-Sided Ideals of Rings of Matrices

This is a question I posted in this topic: Pathologies in "rng". However, I decided that the question deserves its own thread, and I want to know if anybody can answer it. For ...
0
votes
2answers
37 views

List Elements of Left/Right Cosets of $H$ for: $G=\mathbb{Z}_{15}, H=\langle 5 \rangle $

"A Book of Abstract Algebra" presents this exercise: In each of the following, $H$ is a subgroup of $G$. List the cosets of $H$. For each coset, list the elements of the coset. ...
0
votes
2answers
39 views

Find the Cosets of Subgroup $\langle 3 \rangle $ of $\mathbb{Z}$

"A Book of Abstract Algebra" presents the following exercise: Describe the cosets of the subgroups: The subgroup $\langle 3 \rangle $ of $\mathbb{Z}$ Given that $H=\langle 3 \rangle $, I ...
2
votes
0answers
51 views

Composition factoring into nonnegative integer polynomials

Consider the integer polynomials with nonnegative coefficients, such as: $$ 1 + 2x +2x^2 $$ $$ 3 + 3x^4 + 11x^{10}$$ $$ 13 + 7x + x^2 $$ I asm interested in knowing "what is a composition ...
2
votes
0answers
12 views

Find the intertwiner of two equivalent group representations

Let $G$ be a finite group. Let $$\{A(g)\mid g \in G\} \text{ and }\ \ \ \{B(g) \mid g \in G\}$$ be two equivalent unitary matrix representations of $G$. How do I find a unitary intertwining matrix ...
1
vote
1answer
36 views

Confusion about the centre of a p-group

If a non-cyclic group $G$, non-commutative also has order $p^{3}$ does that mean for every $x\in G$ , $x^{p}$ is in $Z(G)$? I am trying to solve a problem from $p$-groups and at this point I ...
0
votes
2answers
25 views

Field $F$ with $\operatorname{char}F=3$ and algebraic over $\mathbb{F}_3$ has a primitive root of unity.

Suppose that $F$ is a field with $\operatorname{char}F=3$ and $F$ is an algebraic extension of $\mathbb{F}_3$. Prove that $F$ contains a primitive $n$th root of unity for some $n>2$.
1
vote
2answers
60 views

Why do nil ideals annihilate simple modules?

A nil ideal $N$ of a ring $R$ is defined as follows: $(N,+)$ is a subgroup of $(R,+)$ $\forall x \in N, \forall r \in R :\quad x \cdot r \in N$ $\forall x \in N, \forall r \in R : \quad r \cdot x ...
0
votes
1answer
36 views

The characteristic of real-closed fields is zero?

We know that $F$ is a real-closed field if $F$ is not algebraically closed but $F(\sqrt{-1})$ is algebraically closed. So I have this question What can we say about $\operatorname{char}F$? Is it ...
7
votes
3answers
1k views

Can an ordered field be finite?

I came across this question in a calculus book. Is it possible to prove that an ordered field must be infinite? Also - does this mean that there is only one such field? Thanks
0
votes
0answers
33 views

Projective and injective resolutions of cyclic modules

Let $R$ be a ring and $M$ a cyclic $R$-module. It is well-known that always exist projective and injective resolutions of $M$. Is it any method to construct explicitly these resolutions which have the ...
2
votes
1answer
22 views

Decomposition of Torsion Module

Let $k$ be a field, $k[X]$ the polynomial ring in one variable and $M$ a torsion $k[X]$-module (not necessarily finitely generated). Consider the submodules \begin{equation*} M_1 = \{a \in M \mid ...
2
votes
2answers
41 views

$K/F$ Galois Extension, $H \leq G$, where $G$ the Galois group, then there is $\alpha \in K$ s.t. $H=\{\sigma \in G : \sigma \alpha = \alpha\}$.

Let $K/F$ be a finite Galois extension of fields with Galois group $G$. Let $H$ be a subgroup of $G$. Then there is $\alpha \in K$ such that $H=\{\sigma \in G : \sigma \alpha = \alpha\}$. My proof ...
-1
votes
1answer
43 views

a question of group theory [duplicate]

let $S$ be a collection of (isomorphism classes of) group $G$ which have the property that every element of $G$ commutes only with the identity element and itself then which option is true and why ? ...
0
votes
0answers
20 views

$ G $ is supersoluble if and only if every normal subgroup of $ G $ with index no more than 2 satisfies the maximal permutizer condition.

Permutizer of a subgroup $ H $ of $ G $ is defined to be the subgroup generated by all cyclic subgroups of $ G $ that permute with $ H $, i.e. $ \langle x\in G \vert \langle x \rangle H = H \langle ...
0
votes
2answers
27 views

Interpreting a Quotient Group ($D_8/\langle r^2\rangle$) in 2 Distinct Ways

I seem to have a misunderstanding about quotient groups. Let $D_{2n}$ denote the group of symmetries of an $n$-gon and let $V_4$ denote the Klein-4 group. On one hand, if we identify $r^2$ with $1$, ...
1
vote
1answer
53 views

Why the extension dimension of $x^3-2$ equal to $6$?

I have seen couple questions related to this one, but after reading the answers I am still confused: Why is the extension dimension of $x^3 - 2$ equal to $6$? In other words, why are the basis ...
-1
votes
1answer
48 views

True or false simple algebra questions (centralizers, conjugacy classes, normal groups, abelian groups) [closed]

Can someone please verify my answers to the following questions? Note: This is NOT homework! Answer true or false to the following questions: Two elements of a group in the same conjugacy ...
1
vote
1answer
65 views

I have to show $A$ is not cyclic

Suppose that $A=C\oplus C=\begin{pmatrix} C & 0 \\ 0& C \end{pmatrix}$, $C$ be a companion matrix of $m(x)=m_0+m_1x+\ldots+m_{n-1}x^{n-1}+x^n$ . I have to show that $A$ is not cyclic. Can any ...
4
votes
3answers
123 views

Every normal subgroup is the kernel of some homomorphism

Clearly the kernel of a group homomorphism is normal, but I often hear my professor mention that any normal subgroup is the kernel of some homomorphism. This feels correct but isn't entirely obvious ...
3
votes
1answer
65 views

Finding the Fixed Fields in the Galois Correspondence for the Splitting Field of $x^4-3x^2+4$ over $\mathbb{Q}$

I have found the Galois group of the polynomial $x^4 - 3x^2 + 4$ (see below), but I am not sure how to find the fixed fields in the Galois correspondence. The roots of the polynomial are $$\pm ...
1
vote
1answer
20 views

Notation involving field extension

I am currently reading notes on Galois Theory and have come upon the following proposition, Let $f(x)$ be the minimal polynomial of a generator $\alpha$ of a finite field extension $k(\alpha)$ of ...
1
vote
1answer
30 views

Relating definitions of a normal field extension.

I have come across the following two definitions of a normal field extension. $\textbf{Definition 1:}$ An algebraic field extension $L/K$ is said to be normal if $L$ is the splitting field of a ...
2
votes
1answer
52 views

Finding all ideals in a finite ring

Let $\mathbb F_2$ be the field of two elements. Consider the factor ring $$R=\mathbb F_2[x, y]/\langle x^2, y^2\rangle.$$ I want to find all ideals of $R$. Note that ...
0
votes
1answer
32 views

If $F$ is a field, then any two algebraic closures are isomorphic by an isomorphism that is the identity on $F$.

To start, suppose $K_1$ and $K_2$ are two algebraic closures of $F$. (a) Let $P$ be the set of partial functions $f$ from $K_1$ to $K_2$ with the following properties: $F$ is contained in ...
1
vote
0answers
30 views

Dihedral group and symmetry group of icosahedron

I am studying abstract algebra at the moment, but I have several troubles picturing the dihedral group $D_n$ and the symmetric group of the icosahedron. For instance, I find it really hard to solve ...
1
vote
0answers
22 views

$ G $ is satisfies the maximal permutizer condition if $ G $ is supersoluble?

Permutizer of a subgroup $ H $ of $ G $ is defined to be the subgroup generated by all cyclic subgroups of $ G $ that permute with $ H $, i.e. $ \langle x\in G \vert \langle x \rangle H = H \langle ...
0
votes
1answer
24 views

Semidirect Product of Two Groups

So I am beginning to learn Semidirect product. Now I have to identify the semidirect product of the two groups $Z_{p}\times Z_{p}$ =$H$ and $K$=$Z_{p}$ where p is an odd prime. So I can write ...
5
votes
1answer
104 views

Problem book on linear algebra

Please refer a problem book on linear algebra containing the following topics: Vector spaces, linear dependence of vectors, basis, dimension, linear transformations, matrix representation with ...
1
vote
1answer
25 views

Prove socle is ideal

In any ring $R$ define the socle as the sum of all minimal right ideals of $R$. Say we have two minimal ideals $A,B$. If $a\in A,b\in B$, then $a+b$ is in the socle. If $x\in R$, then ...
2
votes
2answers
61 views

Ideal of polynomials vanishing on $\{(x,y): x^2+y^2=1, x \neq 0 \}$

I'm reading the book "Introduction to algebraic geometry" by Hassett, and in Chapter 3, after introducing the concept of the ideal of polynomials vanishing on a set $S$, the author gives some ...
0
votes
0answers
24 views

Question on primitive element for extensions of finite fields [duplicate]

I was given this question in field theory class which I am stumped on as after some effort I still cannot tackle. I am given a general natural number $n>1$ and am asked to prove or disprove: ...
2
votes
1answer
33 views

Another abstract algebra/field theory question

Suppose that $F$ is a field, $S \subseteq F^n$ and $I$ is an ideal in $F[x_1, \cdots, x_n] = F[\bar{x}]$. Define $$I(S) = \{ f \in F[\bar{x}]: f(\bar{s}) = 0, \forall \bar{s} \in S\}$$ and ...
5
votes
1answer
56 views

Why restriction to $B(\alpha)$ is a homomorphism from $Gal(E/B)$ to a group with kernel $Gal(E/B(\alpha))$?

I'm reading Galois Theory for Beginners by John Stillwell. It's a good introduction, giving the essence of the idea with minimum algebra complexity. However, I'm a bit lost at his Theorem 2 (the ...
2
votes
1answer
54 views

Number of abelian groups of order 108 [duplicate]

What is the number of abelian groups of order 108 upto isomorphism ? To answer this I wrote explicitly the possible abelian groups of order 108 as follows : $$\Bbb Z_{108}$$ $$\Bbb ...
1
vote
2answers
54 views

Cardinality of a ring obtained by quotienting $\Bbb Z[x]$

Let $R$ be the ring $\Bbb Z[x]/((x^2+x+1)(x^3 +x+1))$ and I be the ideal generated by $2$ in $R$. What is the cardinality of the ring $R/I$? 27 32 64 infinite Now I was thinking $R$ could be ...
14
votes
2answers
174 views

Geometric reason as to why $H^2$ of the Klein bottle is $\mathbb{Z}/2\mathbb{Z}$?

I was reading this document when I came across the following: Recall that $H^2(K; \mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}$. Here $K$ denotes the Klein bottle. Is there a good geometric ...
-1
votes
1answer
42 views

Find the number of units in $M_2(\mathbb Z_p)$ where $p$ is prime. [duplicate]

Find the number of units in $M_2(\mathbb Z_p)$ where $p$ is prime. I want a detailed solution, not just the number. $M_2$ means matrix of order $2\times 2$. I know the defn of units. But how to ...
2
votes
1answer
29 views

idelic ray class group modulo $\mathfrak{m}$

I'm studying the idele group $\mathcal{I}$ for a number field $K$. My definition of the ray class group attached to a modulus $\mathfrak{m}$ is $$\mathcal{C}_{\mathfrak{m}}= ...
1
vote
1answer
45 views

Infinite tower of algebraic extensions

For all I know, the following fact should be true: Consider an infinite tower of extensions $L_0 \subset L_1 \subset L_2 \subset \cdots$ such that $L_{i + 1} / L_i$ is algebraic for all $i \in ...
1
vote
3answers
28 views

Centre of matrix ring over skew field

Let $R$ be a semisimple ring. Show that $R$ is simple iff the centre of $R$ is a field. Book's solution: If $R$ is simple, it has the form $\mathfrak{M}_n(K)$ for a skew field $K$, and its ...
3
votes
0answers
35 views

Finding relations of variables

Suppose that \begin{align*} x&=t+t^{-1}+t^2s+t^{-2}s^{-1}+ts^{-1}+t^{-1}s-6\\ y&=t+t^{-2}+ts+s^{-1}-4\\ z&=t^{-1}+t^2+t^{-1}s^{-1}+s-4 \end{align*} Find a polynomial $P(x, y, z)=0$ ...