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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If $R[X]$ is ED then $R[X] $ is PID

Is this true and why. If $R[X]$ is ED then $R[X] $ is PID . Thanks for help.
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22 views

Exact functors preserve free modules?

Let $R$ be a principl ideal domain. Suppose that $F$ be an exact functor from the category of $R$-modules to itself. If $M$ is a free $R$-module, is $F(M)$ still a free $R$-module? I do not know how ...
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11 views

Order of prime ideals over split primes in the class group.

Let $P$ be a prime ideal of $\mathcal O_K$ ($K$ a quadratic field) and let $P$ have norm $p$ where $p$ is a split prime. Is it possible for the ideal class $[P]$ to have order less than three? I feel ...
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42 views

Multiplicative Inverse Element in $\mathbb{Q}[\sqrt[3]{2}]$

So elements of this ring look like $$a+b\sqrt[3]{2}+c\sqrt[3]{4}$$ If I want to find the multiplicative inverse element for the above general element, then i'm trying to find $x,y,z\in\mathbb{Q}$ such ...
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1answer
19 views

Prove this result relating to the sign of a permutation

Suppose that $\phi \in S_n$ is a permutation. Suppose also that $\psi = \phi \circ (i,j),$ where $1 \leq i, j \leq n.$ Why does it follow that sign$(\phi) = $ $-$sign$(\psi)$?
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44 views

Show that If $R[X]$ is Euclidean domain then $R$ is a field

Let $R$ is an integral domain . Show that If $R[X]$ is Euclidean domain then $R$ is a field . I'll be waiting for your help. Thank you very much in advance!
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1answer
18 views

Noetherian ring of symmetric polynomials

I wish to show that $k[x_1,x_2,..,x_n]^{\Sigma_n}$, which is the ring of all symmetric polynomials, is Noetherian. I thought the easiest way to do this would be to show that every ideal is ...
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11 views

If $K$ is a complex of $R-$modules and $J$ is an injective $R-$module, prove that $H^n(Hom_R(K,J))\cong Hom_R(H_n(K),J)$

If $K$ is a complex of $R-$modules and $J$ is an injective $R-$module, prove that \begin{equation*} H^n(Hom_R(K,J))\cong Hom_R(H_n(K),J). \end{equation*} Thank you in advance.
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Is the congruence relations lattice of a lattice a sublattice of all equivalence relations on it?

By this Wikipedia link, it seem the set of all congruence relaions on a lattice $(X,\le)$ is a complete lattice with inclusion. Is this lattice a (complete) sublattice of the lattice of all ...
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73 views

Verifying if a multiplication table is from a group

I'm asked to verify which of these multiplication tables form a group. I'm having problems to see which of the axioms for a group are violated in each table. In (a), I couldn't find an element $e$ ...
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28 views

How to find the Galois group of a given polynomial from Galois viewpoint?

Excerpt from Page 3 Consider the polynomial equation \begin{equation*} x^4 - 2 = 0, \end{equation*} with integer coefficients. We have four solutions: \begin{equation*} \alpha = ...
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4answers
57 views

Is this a valid way of solving modular equations?

Some of the exercises in my abstract algebra textbook involve solving congruence equations, for example $$5x \equiv 1 \pmod 6$$ I convert that to a linear equation by rewriting $$x = \frac{k6+1}{5} = ...
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0answers
15 views

Reference Request: Replacement for Anderson and Fuller

I'm working through Rings and Categories of Modules by Anderson and Fuller. It's a great text, but I would like a more modern replacement. The text focuses too much on the language of generation and ...
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2answers
38 views

Problems related to field theory [on hold]

Suppose that $ F $ is a field whose characteristic is not 2. If nonzero elements of $ F $ form a cyclic group under multiplication then show that $F $ is finite
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67 views

Structure of the group $\{1+p\mathbb Z_p \}$

In preparation for algebraic number theory I am reading Serre : A course in Arithmetic. I stuck in understanding a proof (p.17): Notation: $U_n=1+p^n\mathbb Z_p$ Actually there are many things ...
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1answer
50 views

Show every $a \in E^*$ is a root of $x^{p^d-1} -1 $?

Let $\mathbb{Z}_p < E$ be an extension field of degree $d$. A simple counting argument shows: $|E^*| = p^d - 1$ Proposition: For all $\alpha \in E^*$, $x^{p^d-1} -1 = 0.$ In a field of $p^d ...
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1answer
26 views

Quadratic residue and permutation [on hold]

Let $p>2$ be a prime number and $a \in \mathbb{Z}_{p}$. For an integer $k$ consider the permutation $\pi$ of the set defined by $\pi: n \to kn+a \pmod p$. Prove that $k$ is quadratic residue modulo ...
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14 views

Etingof problem 2.15.1 Representations of sl(2)

I'm studying from Etingof's Introduction to Representation Theory. This is problem 2.15.1, part a. I feel I'm close to the solution. Here's what I have. Problem: A representation of sl(2) is a vector ...
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2answers
35 views

Generator of group $D_n$

Let be $D_n$ the group of symmetries of a regular n-sided polygon. Prove that $D_n$ is generated by a minimum rotation angle and a reflection (about a symmetry axis). I really do not know how to ...
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29 views

Bijection and field [on hold]

How to prove that $\phi : \mathbb{Z}_{pq} \to \mathbb{Z}_{p} \times \mathbb{Z}_{q} $ such that $\phi (x) = (x_p , x_q) $ where $x_p = x$ (mod $p$) and $x_q = x$ (mod $q$) is a bijection?
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1answer
19 views

Give an extension field of $\mathbb{Z}_3$ of degree 3?

I have an irreducible polynomial in $\mathbb{Z} $ That irreduible polynomial is: $(1)$ $x^3 + 2x + 1$ I know that this polynomial creates a maximal ideal and that I can create an extension field ...
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3answers
50 views

Finding the fixed Field of $\sigma \in Aut(\mathbb{R}(t)/\mathbb{R})$

Finding the fixed Field of $\sigma \in Aut(\mathbb{R}(t)/\mathbb{R})$ Let $\sigma$ be such that $\sigma(t)=-t$. I assume there is only one automorphism like this, I am not sure exactly why... How ...
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21 views

Product of any two non disjoint cycles

Suppose you have a permutation $\sigma = \sigma_1 \sigma_2$ where $\sigma_1 = (i_1 i_2...i_k)$, $\sigma_2 = (j_1 j_2...j_l)$ and $i_{k_1},i_{k_2},...,i_{k_r}$ are equal to ...
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3answers
84 views

Example of commutative ring that doesn't satisfy distribution of intersection over addition

I'm trying to find an example of commutative ring $R$ and ideals $\mathfrak a,\mathfrak b,\mathfrak c \in R$ such that $$\mathfrak a \cap (\mathfrak b + \mathfrak c) \neq \mathfrak a \cap ...
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4answers
172 views

Fields of arbitrary cardinality

So I took an introductory abstract algebra course a few semesters ago, and we were shown that groups and rings can both be made into products, i.e. if I have some group $G$ (resp. ring $R$) and some ...
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1answer
23 views

Ring with no identity (that has a subring with identity) has zero divisors.

Let $L$ be a non-trivial subring with identity of a ring $R$. Prove that if $R$ has no identity, then $R$ has zero divisors. So I assumed that there $\exists$ $e \in L$, such that $ex=xe=x$, ...
3
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1answer
29 views

$a,b,N$ are integers. Prove $x=x_0+\cdots$, $\ \ y=y_0+\cdots $ are solutions to $ax+by=N$

I'm asked to prove that if $a,b,N$ are integers, then in the equation: $$ax+by=N$$ I must prove that the integers $$x=x_0+\frac{b}{d}t,\ y=y_0-\frac{a}{d}t$$ are solutions to the equation. where ...
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743 views

If I know the order of every element in a group, do I know the group?

Suppose $G$ is a finite group and I know for every $k \leq |G|$ that exactly $n_k$ elements in $G$ have order $k$. Do I know what the group is? Is there a counterexample where two groups $G$ and $H$ ...
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0answers
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I.N. Herstein , Topics in Algebra problem 2.5.24 [duplicate]

Let $G$ be a finite group whose order is not divisible by 3. Suppose that $(ab)^3=a^3b^3$ for all $a,b\in G$. Prove that $G$ must be abelian.
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112 views

What is mathematical structure?

When we have an isomorphism, between 2 groups or vector spaces let us say, then it is said to be structure preserving. An isomorphism exists when there is at least one mutually invertible morphism ...
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1answer
38 views

Show that $|\operatorname{Aut}(\mathbb{Q}(\sqrt[10]2))|=2$

Let $\mathbb{K}=\mathbb{Q}(\sqrt[10]2)$. Show that $|\operatorname{Aut}(\mathbb{Q}(\sqrt[10]2))|=2$. Well, it is easy to see that the degree of this extension over $\mathbb{Q}$ is ten. Also, is ...
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The group $(1+p\mathbb Z_p)/(1+p^{n}\mathbb Z_p)$

I want know some information about the group \begin{equation*} \frac{(1+p\mathbb Z_p)}{(1+p^{n}\mathbb Z_p)} \end{equation*} (the Quotient group). What is the order of this group? I guess $p^{n-1}$ ...
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1answer
54 views

Wrong proposition in “Atiyah and Macdonald”s book?!

In page 6 of "Introduction to commutative algebra" says that: $a \cap b = ab$ provided $a + b = (1)$ But i think it's not true,by considering $a = b = (2) \in \mathbb Z_6$
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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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2answers
58 views

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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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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27 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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21 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 ...
3
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1answer
21 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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2answers
35 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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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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43 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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45 views

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 seen something like this in my course, so I'm a bit skeptical, since this seems like a very simple way of computing norms of ideals. If ...
2
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1answer
24 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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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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47 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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2answers
45 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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30 views

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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1answer
16 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 ...