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.

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Generalize a result to any category.

Consider two categories $\mathscr{C}$ and $\mathscr{D}$ where $\mathscr{C}= Grp$ and $\mathscr{D}= \textbf{Set}$, then we are taking the forgetful and faithful functor $p$ (this is, we have a group ...
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Show $R/I$ is a ring with unity,$1 + I$ [on hold]

Suppose $R$ is a ring with unity and $I \neq R $ is an ideal of $R$. Show that $R/I$ is a ring with unity,$1 + I$ . Can anyone give me a hit to do this question? Thanks
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Are there differences between the International and U.S. Editions of Dummit and Foote?

I'm taking an abstract algebra course this fall and want the textbook over the summer. I found an international copy of Dummit and Foote Abstract Algebra 3rd ed. for much cheaper than the U.S. ...
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Let $G$ be an abelian group of order $(p^n)m$, where $p$ is a prime and $m$ is not divide by $p$.

Let $P = \{a\in G\mid a^{p^k} = e\text{ for some $k$ depending on $a$}\}$. Prove that (a) $P$ is a subgroup of $G$. (b) $G/P$ has no elements of order $p$. (c) order of $P=p^n$.
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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 ...
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If a Bilinear Form is Non-Degenerate 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 ...
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A property of finite field of order $2^n$

Suppose $a$ and $b$ are elements of a finite field of order $2^n$ with $n$ odd and $a^2+ab+b^2=0$. Is it necessary that both $a$ and $b$ must be zero ? I understand that the field has characteristic ...
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31 views

Grothendieck Group of a commutative group

I know that the Grothendieck group of a commutative monoid is $G(M) = M XM/R$ where $(x,y)R(x',y') $ iff$ $ there is a $z$ in $M$ such as $ x*y'*z=y*x'*z$ , $*$ is the law of the monoid. My ...
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A Theorem on Skew-Symmetric Bilinear Forms in Hoffman and Kunze's Book

On pg. 377 in Hoffman and Kunze's Linear Algebra(Second Edition) Theorem 6 reads: Let $V$ be an $n$-dimensional vector space over a subfield of the field of complex numbers, and $f$ be a skew ...
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2answers
60 views

If $\dim_F F[A] <n$, then $A$ is not cyclic. [closed]

Suppose that $A \in M_n(F)$ and the minimal polynomial of $A$ is irreducible. S0 $F[A]$ is a field extension of $F$. I have to show: 1) If $\dim_F F[A] <n$, then $A$ is not cyclic. 2) If $\dim_F ...
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Can a field extension still have “non-separability” above its maximal purely inseparable subextension?

Question 1 Let $E/F$ be an algebraic field extension. Let $K$ be the set of all elements of $E$ that are purely inseparable over $F$. Then, $E/K/F$ is a tower of fields, and $K/F$ is purely ...
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3answers
93 views

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

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 ...
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38 views

$f'(x) \equiv 0 \pmod{p}$ with $\deg f < p$ implies $f(x) \equiv c \pmod{p}$

Let $f(x) = P(x)/Q(x)$ where $P, Q \in \mathbb{Z}[x]$. Define $\deg f = \max(\deg P, \deg Q)$. Then as usual, $f'(x) = (Q(x)P'(x) - P(x)Q'(x))/Q(x)^2$. Suppose for some prime $p$, we had $f'(n) ...
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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 ...
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Do all n x n matrices over the reals represent linear transformations?

Do all $v \in M_n (\mathbb{R})$ represent linear transformations? To add to that a bit to further clarify for myself: Looking up the def. of a transformation it is any function $f$ mapping a set $X$ ...
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80 views

Roots of $f(x) = x^3+x^2-2x-1$

Roots of $f(x) = x^3+x^2-2x-1$ Show: $a_1=2\cos(\frac{2\pi}{7})$ is a root of $f$. [Edited here] $a_2 = a_1^2-2$ is a root of $f$. $a_3 = a_1^3-3a_1$ is a root of $f$. The first one is ...
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49 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*} ...
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1answer
39 views

About right identity which is not left identity in a ring

Let $S$ be the subset of $M_2(\mathbb{R})$ consisting of all matrices of the form $\begin{pmatrix} a & a \\ b & b \end{pmatrix}$ The matrix $\begin{pmatrix} x & x \\ y & y ...
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Prime 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 ...
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1answer
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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 ...
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2answers
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Faithful module is infinitely dimensional as a 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 ...
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Why is not the ring $\mathbb{Z}[2\sqrt{2}]$ a unique factorization domain?

Why is not the polynomial ring $R[x]$ a unique factorization domain, where $R$ is the quadratic integer ring $\mathbb{Z}[2\sqrt{2}]$? I'm trying to find a irreducible nonprime element or ...
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1answer
89 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 $\mathbb{Q}(\pi) \cong \mathbb{Q}[[x]]/ \langle \sin(x) \rangle$?
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graded Hopf algebra and its dual

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

Transcendence bases [closed]

Let $\Bbb k \subset \mathbb{K}$ be a field extension. Let $S_1,S_2 \subset \mathbb{K}$ be sets of algebraically dependent elements such every proper subset of $S_i$ is algebraically independent. I ...
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4answers
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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 ...
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$I$ 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 ...
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Relation between hyperbolic numbers and hyperbolic functions

Is there any relation between the hyperbolic (split-complex) numbers and hyperbolic trig functions? Or are they just named similarly by accident?
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55 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 ...
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Is $\ker(nat_{H})=H$ a true statement? [closed]

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 $\ker(nat_{H})$.
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1answer
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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 ...
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Divisibility of polynomials in a subfield of a field.

I am trying to prove the following assertion: Let $K\subset L$ be fields, let $f,g\in K[x]$ be such that $f\mid g $ in $L[x]$, then $f\mid g$ in $K[x]$. We clearly have that $fh=g$ for some ...
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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 ...
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1answer
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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 ...
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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 $N$ ...
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Proving that the intersection of two subrings of R is also a subring of R

If $R_1$ and $R_2$ are both subrings of $R$ , how to prove that $R_1 \cap R_2$ is also a subring of $R$. here is my attempt (1) since $R_1$ is a subring of $R$ then it must contain zero (identity ...
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Algebra + Real Analysis video lectures

I'm an undergraduate taking graduate courses beginning a research project. I don't have much time but want to brush up on my Algebra and real analysis at a graduate level. Does anybody know any good ...
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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 ...
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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 ...
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Prove that these are not pairwise isomorphic

Prove $\mathbb Z_s$, $G_s$ (the group of symmetries of the square) and the quaternion group $Q$ are not pairwise isomorphic. How would you go about proving. Seems quite difficult. I know that none of ...
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Confused about transcendental numbers [closed]

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 ...
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About the elements of Dihedral Group.

I have some difficulties finding the elements of Dihedral Group $D_8$ (note that the order $|D_8|=8$). I know the geometric approach for defining $D_8$, but I prefer the algebraic way. In this case, ...
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$g$ has order $n$, then $\langle g\rangle=\langle g^2\rangle=\cdots=\langle g^{n-1}\rangle$

If $g$ has order $n$, then $\langle g\rangle=\langle g^2\rangle=\cdots=\langle g^{n-1}\rangle$. This should be fairly easy but somehow I just couldn't prove it. I only managed to prove the case ...
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1answer
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Why not isomorphic?

Need to show why $(S_7,\circ)$ is not isomorphic to $(\Bbb Z/100\Bbb Z,+)$. I think it might have something to do with Abelian, but I'm not sure.
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1answer
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If $|G| = pqr$ for $p<q<r$ primes and all the Sylow groups are normal; is $G$ abelian? [closed]

Let $G$ be a group with $|G| = pqr$ for distinct primes $p<q<r$. If every Sylow subgroup of $G$ is normal, then is $G$ Abelian? Thank you in advance.
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2answers
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Is group $G$ must abelian, when some condition is given by using exact sequence?

Suppose we are given the following exact sequence of groups where $A$ is an abelian normal subgroup of $G$: $$1 \rightarrow A \rightarrow G \rightarrow Q \rightarrow 1\tag{E}$$ If $G$ is Abelian, ...
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DVR, power series expansion.

Let $A$ be a discrete valuation ring with quotient field $K$, maximal ideal $\mathfrak{m}$, uniformizing parameter $t$. Let $k = A/\mathfrak{m}$, so $k$ is a field. How do I show that there is a ...
2
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1answer
62 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 ...
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The characteristic and minimal polynomial of a companion matrix

The companion matrix of a monic polynomial in 1 variable over a field $F$ plays an important role in understanding the structure of finite dimensional $F[x]$-modules. It is an important fact that the ...