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)

0
votes
2answers
25 views

Simple questions about a polynomial ring

Reading Pinter's algebra, I'm little bit confused. In ch.24, the author says that x which appears in a polynomial is to be considered as a 'placeholder' for a moment... All right, then i was trying ...
1
vote
1answer
40 views

Is it possible that $1\otimes 1 = 0$?

Let $R$ be a commutative ring. Let $A,B$ be $R$-algebras and consider their product $A\otimes_R B$. Is it possible that $1\otimes 1=0$? What is an example? If $R$ is a field, $1\otimes 1$ is never ...
-1
votes
0answers
15 views

Is $\ker(nat_{H})=H$ a true statement? [on hold]

Is $\ker(nat_{H})=H$? $nat_{H}$ defines as $nat_{H}(a)=a*H$ I know what $ker$ and $nat_{H}$ are but I am not familiar with $\ker(nat_{H})$.
1
vote
1answer
15 views

If a Bilinear Form is Non-Degenerte on a Subspace $W$, then $V=W\oplus W^\perp$.

$\newcommand{\range}{\text{image}}\newcommand{\ann}{\text{Ann}}\newcommand{\set}[1]{\{#1\}}$ Problem: Let $V$ be a finite dimensional vector space over a field $F$ and $f$ be a symmetric bilinear ...
0
votes
1answer
24 views

Subgroups of automorphisms of Finite fields

Let $G$ denote the group of all the automorphisms of the Field $F_{3^{100}}$.Then,what is the number of distinct subgroups of $G$? First of all I have to compute $G$. Now \begin{equation*} ...
2
votes
1answer
18 views

Are binomial series multiplicative in their bases?

In $ℚ[Z]$, by $Z \choose k$ denote the polynomial $${Z \choose k} = \frac{1}{k!}·\prod_{i=0}^{k-1} (Z-i),$$ so that ${Z \choose k}(n) = {n \choose k} = \frac{n!}{k!·(n-k)!}$ in $ℚ$. Now, in the ring ...
0
votes
0answers
23 views

How is the second part of a dual number called?

A complex number $a + bi$ has a real part $a$ and an imaginary part $b$. But, what about dual numbers $u + v\epsilon $? I have seen the non-real part $v$ been called the infinitesimal part. Is this a ...
1
vote
1answer
25 views

$I$ is a two sided ideal in $A$, $L$ and $M$ are $A$-modules such that $L \subset M$. If $l\in L$ and $l\in MI$ then $l \in LI$?

$A$ is an Artinian ring, $I$ is a two sided ideal. I know that $L \subset M$ are $A$-modules, and that $l \in L \cap MI$. Is it true that $l \in LI$? If is not true, can I have an example where it ...
0
votes
1answer
27 views

If N is a submodule of M, then λ(M) = λ(N) + λ(M/N).

Defining length of a module as follows, a module M has length λ(M) = n if there is a chain of submodules 0 = $M_{0}$ < $M_{1}$ < · · · < $M_{n}$ = M where n is maximal, and λ(M) = ∞ if there ...
1
vote
1answer
34 views

When does $ \langle gI, t \rangle = \langle I, g^{-1} t\rangle $ hold true?

Consider $I, t \in \mathbb{R}^d$ and $g$ is some element in a group of transformations (for example like the affine group in $\mathbb{R}^2$). I was wondering when the inner product $ \langle gI, t ...
1
vote
3answers
48 views

Determining whether or not an element is integral over $\mathbb Z$

I want to prove that $\alpha=\frac{\sqrt[3]{2}}{2}$ is not integral over $\mathbb{Z}$ (i.e. not an algebraic integer). Is there a way to modify the following reasoning to make it work? A quick check ...
1
vote
1answer
30 views

How is a symmetric group the subgroup of the group of isometries of three-dimensional space?

So I have this question to solve. I've already shown that the group of rotations of a cube is isomorphic to $S_4$. I need to prove that these two groups are not conjugate when considered as subgroups ...
-1
votes
3answers
59 views

Confused about transcendental numbers [on hold]

I'm little confused about the type of numbers that had been known, for example, consider a polynomial equation with rational and irrational coefficients of a degree p-prime number that is greater than ...
2
votes
1answer
77 views

Is $\mathbb{Q}(\pi) \cong \mathbb{Q}[[x]]/ \langle \sin(x) \rangle$?

If we let $\mathbb{Q}[[x]]$ be the set of all power series with rational coefficients then can we say that $\mathbb{Q}(\pi) \cong \mathbb{Q}[[x]]/ \langle \sin(x) \rangle$?
2
votes
0answers
17 views

graded Hopf algebra and its dual

I am learning Hopf algebra, and there are two questions as follows: Is the tensor product of two hopf algebras still a Hopf algebra? Let A be a infinite dimensional algebra. Is the dual A* a ...
2
votes
1answer
61 views

Group presentation of Integers $\big(\mathbb{Z,+}\big)$

I can't understand how is it possible to represent the group $(\mathbb{Z},+)$ as follows $$\mathbb{Z} = \big<a\big>$$ with only one generator and no relations ? How can there be no relations ...
-5
votes
0answers
60 views

What are $S_{n}$ and $A_{n}$ in group theory? [on hold]

What are $S_{n}$ and $A_{n}$ in group theory, and is $[S_{4},A_{4}]=4$? I know that $S$ has to do with permutations, but I am not sure if thats right. Thanks,
1
vote
1answer
30 views

Abstract Algebra: Proof involving a p-Sylow, nomal subgroup P in G. [on hold]

Let $P$ be a normal subgroup in $G$ where $P$ is a $p$-Sylow subgroup of $G$. Show that $\phi(P)=P$ for every automorphism $\phi$ of $G$.
-2
votes
1answer
36 views

Homomorphisms between abelian groups [on hold]

Is the following a true statement? Let $H$ and $K$ be finite Abelian groups. Assume that $H$ has order $n$ and $K$ has order $m$. If $n$ and $m$ are relatively prime, then every homomorphism $\alpha ...
0
votes
2answers
32 views

If $M\bigoplus N $ submodule $A\bigoplus B$ does it imply either $M$ submodule $A$ or $M$ submodule $ B$

If $M\bigoplus N$ is a submodule of $A\bigoplus B$ does it necessarily imply either $M$ is a submodule of $A$ or $M$ is a submodule of $B$. If in general it is not true, is it true if $M$ and $N$ ...
-1
votes
0answers
15 views

does dicyclic group of order $2^{\alpha}m$, $(2,m)=1$ have normal subgroup of order $m$?

consider dicyclic group with presentation $$Dic_n=< a,b |a^{2n}=1,a^n=b^2,ba=a^{-1}b>$$ the order of this group is $4n$,suppose that $|Dic_n|=2^{\alpha}m$ where $(2,m)=1$ my question is about ...
3
votes
2answers
56 views

Is $\mathbb{Z}[2\sqrt{2}]$ a PID?

I am practicing for my algebra qual and I would like to know if $\mathbb{Z}[2\sqrt{2}]$ is a PID. I had no intuition at first except the fact that $\mathbb{Z}[i\sqrt{2}]$ is a ED with norm ...
1
vote
2answers
26 views

How do I compute the norm of a non-principal ideal of the ring of integers of a quadratic field without using ''large'' results

I am trying to compute the norm of the ideal $I=(7, 1+\sqrt{15}) \trianglelefteq \mathbb Z[\sqrt{15}],$ the ring of integers of $\mathbb Q[\sqrt{15}].$ I knew $I^2$ would be principal, as $I\bar ...
0
votes
4answers
62 views

Consider extension $[\mathbb{Q}(\alpha\ ):\mathbb{Q}]$ where $\alpha\ $ is zero of $P(x) = x^4 + 9x^{2} + 15 $.

Consider extension $[\mathbb{Q}(\alpha):\mathbb{Q}]$ where $\alpha$ is zero of $p(x) = x^4 + 9x^{2} + 15 $. Find $[\mathbb{Q}(\alpha):\mathbb{Q}(\alpha^2 + 3)]$. My attempt: By Eisenstein's ...
0
votes
0answers
19 views

Newton method for $p$-adic fields

I want to understand where the last line comes from. I.e. why there is the $p^{2n-2ka}$ term. I tried to use the estimate formula for the reminder but it doesn't work for me...
4
votes
1answer
40 views

$M$ is a faithful $A$-module.

Let $A = M_2(\mathbb{C})$ and let $M = \mathbb{C}^2$ with its usual $A$-module structure. How do I show that $M$ is a faithful $A$-module? I understand that this amounts to showing that its ...
1
vote
3answers
51 views

Does there exist a module structure over $\mathbb{C}^2$ as a $\mathbb{C}[x]$-module?

Let $R = \mathbb{C}[x]$, $M = \mathbb{C}^2$. Does there exist a module structure over $M$ as an $R$-module or not?
2
votes
2answers
21 views

Module $M$ is infinite dimensional as a $\mathbb{C}$-vector space.

Let $M$ be a $\mathbb{C}[x]$-module. Since $\mathbb{C} \subset \mathbb{C}[x]$, we may consider $M$ as a $\mathbb{C}$-vector space. If $M$ is faithful, must $M$ be infinite dimensional as a ...
-1
votes
1answer
38 views

Determine if $(((13)),\circ)$ is a normal subgroup of $(S_{3},\circ)$ [on hold]

Let $((13))$ denote the group generated by $(13)$. Is $(((13)),\circ)$ a normal subgroup of $(S_{3},\circ)$? Also is $(((123)),\circ)$ a normal subgroup of $(S_{3},\circ)$? I have just started ...
0
votes
1answer
49 views

Equivalence between category of $R$-modules and $S$-modules

Is it true that by equivalence between category of $R$-modules and $S$-modules we conclude that $R$ and $S$ are isomorphic?($R$ and $S$ are unital rings.)
5
votes
0answers
35 views

Exists an $R$-module homomorphism $\varphi: M \to \mathbb{Q}/\mathbb{Z}$ such that $\varphi(m) \neq 0$?

Let $M$ be an $R$-module with some torsion element $m$. Does there exist an $R$-module homomorphism $\varphi: M \to \mathbb{Q}/\mathbb{Z}$ such that $\varphi(m) \neq 0$?
1
vote
1answer
44 views

all of subgroups of group

Is the way to gust that in finite group how many subgroup of same order?I ask this question because when draw the lattice diagram of subgroups of group sure that all of them describe. Thanks for hint
2
votes
1answer
27 views

Arbitrary elements in a quotient ring $\Bbb R[x]/(x-1)$

If I have an ideal $(x-1)$ for the ring $\Bbb R[x]$, how do I think of the quotient ring $\Bbb R[x]/(x-1)$? I have all polynomials with: $$a_nx^n+a_{n-1}x^{n-1}+\cdots+a_2x^2+a_1x+a_0 {\pmod {x-1}}$$ ...
-2
votes
1answer
32 views

Properties of homomorphisms

I have some problems in how to prove these: Let $f$ be homomorphism from group $G$ to a group $N$. Prove the following: $k\le G$ iff $f[k]\le N$ $f$ is onto iff range of $f =N$ $f$ is one-to-one ...
4
votes
0answers
32 views

Heyting algebras and infinite distributive law

I want to prove that "a complete lattice satisfies the infinite distributive law $a\wedge(\vee{S})=\vee\{a\wedge s|s\in S\}$ iff it is a Heyting algebra". I proved "if" part but can't "only if" part. ...
5
votes
1answer
35 views

Group ring of a cyclic group over a finite field

Suppose $ p $ a prime integer and $ n $ a positive integer. Does anyone know off the top of their heads if the group ring $ \mathbb{F}_{p}[\mathbb{Z}/n] $ (perhaps regarding $ \mathbb{Z}/n $ as the $ ...
6
votes
2answers
78 views

Name for the reals augmented with an $x$ such that $x^2 = x$

If you add an $x$ such that $x^2=-1$ to the reals, you get the complex numbers. If you add an $x$ such that $x^2=0$ to the reals, you get the dual numbers. If you add an $x$ such that $x^2=+1$ to the ...
2
votes
4answers
73 views

Prove that $a$ and $a^{-1}$ inverse have the same order in $Z_n$

So there is a question in my lecture notes that I'm not too sure how to approach. It reads as follows: Suppose $a$ is invertible modulo $n$. Prove that $a$ and $a^{-1}$ have the same order in ...
1
vote
1answer
24 views

Field Theory Problem in Beachy's Abstract Algebra involving field extensions and transcendental elements.

Let $\mathbb{F}=\mathbb{K}[u]$ where u is transcendental over $\mathbb{K}$. Show that if $\mathbb{K} \subsetneq \mathbb{E} \subseteq \mathbb{F}$ then u is algebraic over $E$. I'm guessing that I need ...
1
vote
1answer
19 views

Let $K\subseteq F$ be a finite extension of fields, $M$ and $N$ intermediate fields separable on $K$. Show that $MN$ (compositum) is separable.

Let $K\subseteq F$ be a finite extension of fields, $M$ and $N$ intermediate fields separable on $K$. Show that $MN$ (compositum) is separable. Using the Element Primitive Theorem, we know that ...
1
vote
1answer
33 views

Group of order 10 has an element of order 5, without using Cauchy's or Sylow's theorems

This is almost a duplicate of the following questions (but, read further): Group of order $63$ has an element of order $3$, without using Cauchy's or Sylow's theorems Show any group of order ...
-2
votes
1answer
54 views

Let $G$ a finite group with order of $2p$, where $p > 2$ is prime. given that there's $a \in Z(G)$ such that $o(a) = 2$. Prove: $G$ is abelian. [duplicate]

Homework question: Let $G$ a finite group with order of $2p$, where $p > 2$ is prime. given that there's $a \in Z(G)$ such that $o(a) = 2$. Prove: $G$ is abelian. Can you give me some hints ...
1
vote
1answer
35 views

about left identity in a ring..

Let $S$ be the subset of $\mathbb{M}(\mathbb{R})$ consisting if all matrices of the form : $\begin{pmatrix} a & a \\ b & b \end{pmatrix}$ The matrix $\begin{pmatrix} x & x \\ y & y ...
4
votes
0answers
40 views

Ideals in a quotient

I am interested in finding the number of prime ideals in $\mathbb{Z}[x]/(12,x^2+1)$. Here is what I think. Modding out by $x^2+1$, we get $\mathbb{Z}[i]/(12)$. Factoring $12$ in the Gaussian integers, ...
3
votes
2answers
46 views

Understanding Quotient Rings

I am watching a video on Field Extensions (trying to self "relearn" some Abstract Algebra before I take it again). I struggled with it as an undergrad, so I'm trying to get a leg up. The example ...
1
vote
0answers
31 views

A question about a property of Gauss sum.

I am reading the book and I have some questions about Gauss sum. The Gauss sum is defined in the end of page 4, formula (1.14), by \begin{align} g(m,c)=\sum_{a \mod c} \left( \frac{a}{c} \right)_n ...
2
votes
0answers
60 views

Challenging problems in algebra (book recommendation)

Could you suggest me a book/web page where I can find challenging/hard problems in algebra (possibly with solutions) for an undergraduate student (groups, rings, fields, Galois theory)? Thanks in ...
1
vote
0answers
79 views

When does $\frac{\alpha^k-1}{\alpha-1}$ become a unit in $\mathbb{Z}[\alpha]$?

Let $\alpha$ be a complex number. For which $k\in\mathbb{Z}$ does $\frac{\alpha^k-1}{\alpha-1}$ become a unit in $\mathbb{Z}[\alpha]$? If $\alpha=\xi_m$, an $m$-th primitive root of unity, then ...
1
vote
2answers
36 views

How to go about this proof for non zero polynomials.

How do I go about proving this? Let $\mathbb{F}$ be a field and $X$ an indeterminate, and consider the polynomial ring $\mathbb{F}[X]$. Let $f(X), g(X) \in \mathbb{F}[X]$ with $f(X), g(X) \neq ...
1
vote
1answer
32 views

When does it hold that $a^{-1} \in \mathbb{Z}[a]$?

When does it hold that $a^{-1} \in \mathbb{Z}[a]$, for an algebraic number $a $? If $a$ is a root of unity of any order, done. But I know there are other examples: e.g., $2-\sqrt {3}$.