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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Injectivity and surjectivity on algebraic group

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an algebraic group defined over $\bar{\mathbb{F}}_q$, and let $F$ be a ...
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How to workout what elements of a quotient ring look like?

I am trying to understand quotient rings. Firstly: $$\frac{\Bbb Z[x]}{\langle x-1\rangle}$$ The above I can understand in a fairly naive way. Since the ideal is generated by a degree one polynomial, ...
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52 views

An example of a group such that $G \cong G \times G$

I was trying to find an example such that $G \cong G \times G$, but I am not getting anywhere. Obviously no finite group satisfies it. What is such group?
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25 views

Cardinality of Quotient ring

Let $R$ be the ring obtained by taking the quotient of $\mathbb{Z_6}[X]$ by principal ideal $(2x+4)$. Then 1) $R$ has infinite elements 2) $R$ is field 3) $5$ is unit in $R$ 4) $4$ is unit in $R$. ...
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16 views

Annihilators and exact sequence

Let $R$ be a commutative ring and $0\longrightarrow L \overset{f}{\longrightarrow}M\overset{g}{\longrightarrow}N\longrightarrow 0$ be an exact sequence of $R$-modules. How to prove ...
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17 views

The necessary and sufficient condition for a regular n-gon to be constructible by ruler and compass.

I have a problem concerning the necessary and sufficient condition for a regular n-gon to be constructible by ruler and compass. $\bf My$ $\bf question:$ For a given positive integer $n$, how can we ...
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30 views

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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32 views

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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30 views

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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Is there a classification of ideals of $\mathcal O_K$ ($K$ quadratic) over ramified and split primes depending on $d \pmod 4$?

I am unsure if the following argument is correct. I have not been taught to use something like this by my lecturer, so I'm a bit skeptical, since this seems like a very simple way of computing norms ...
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19 views

How to show the existence of an ideal in the ring of Gaussian integers that satisfy the following?

To show that If $p$ is a prime and $p=1\bmod4$ then there exists an ideal of the $R=\mathbb{Z}[i]$, the ring of Gaussian integers, such that $R/I$ is isomorphic to $\mathbb{Z}/p\mathbb{Z}$. I ...
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1answer
27 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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46 views

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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40 views

Is the group $(\Bbb Z,+)$ isomorphic to the the group $(\Bbb Q\setminus\{0\},\cdot)$? [duplicate]

Is the group $(\Bbb Z,+)$ isomorphic to the the group $(\Bbb Q\setminus\{0\},\cdot)$?
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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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14 views

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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76 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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36 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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1answer
26 views

Transcendence bases [on hold]

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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31 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 ...
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48 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? [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 $\ker(nat_{H})$.
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1answer
24 views

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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1answer
40 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
21 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 ...
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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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32 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 ...
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28 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 ...
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37 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 ...
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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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1answer
34 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 ...
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63 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 ...
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1answer
84 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
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 ...
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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,
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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 $G$.
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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 ...
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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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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 ...
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3answers
83 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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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 ...
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68 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 ...
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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...
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41 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 ...
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53 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?
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2answers
24 views

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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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 ...
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1answer
57 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.)