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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Ordered Field and Cauchy-completeness

Let $F$ be an ordered Archimedean Field such that every Cauchy sequence is convergent. I want to prove that if $T$ is an upper bounded subset of $F$, then there exists the supremum of $T$ in $F$. ...
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23 views

extending isomorphism to an automorphism

Lets say $K/F$ is a field extension and $\alpha ,\alpha '\in K$ are two distinct roots of the same irreducible polynomial in $F[x]$. there exists an isomorphism $$\psi:F(\alpha)\rightarrow ...
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13 views

References for loop rings

I recently saw a paper on alternative loop rings, as always, I am interested in all kinds of rings, and new kind of ring looked attractive. I would like to read loop and then loop rings in detail from ...
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$Ext$ and projective dim

I have some problem to understand proof of proposition 8.38 page 473 homology Rotman. Proposition:Let $R^*=R/(x)$,let $x\in Z(R)$ not be a zero-divisor, and let $M$ be a left $R-module$ with $x$ ...
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Galois group of $ \mathbb{Q}(\varepsilon_5) / \mathbb{Q} $

I'm trying to solve the following problem: Let $ L =\mathbb{Q}(\varepsilon_5) $ be an extension of $ \mathbb{Q} $. Find the Galois group $ G(L / \mathbb{Q})$ $ \varepsilon_5 $ is the primitive root ...
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23 views

Question about homomorphisms?

I have a question that asks the following: Let $S,*$ and $T,.$ be binary structures and let the there be a homomorphism betweeen the two. If this is surjective, then if S is a group, so is T. I ...
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40 views

How is $\lbrace a_1, a_2, …, a_n : a_i \in \Bbb Z_2\rbrace$ a group?

I was asked to prove that if we define \begin{equation*} \Bbb Z_2^n = \lbrace a_1, a_2, ..., a_n : a_i \in \Bbb Z_2\rbrace \end{equation*} then it's a group under the operation of addition like ...
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1answer
26 views

Proof of second homomorphism theorem

Let $G$ be a group, $H \leq G$ and $N \unlhd G$. Let $HN=\{hn │h \in H, n \in N\}$. I finished (a) $H \cap N \unlhd H$, (b) $HN \leq G$, (c) $N \subset HN$ and $N \unlhd HN$. I have a question in ...
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25 views

What is the number of elements in $Aut(Q(\pi)/Q)$?

I tried to prove that $|Aut(E/F)|$ is finite, then $E/F$ is a finite extension, but then now I think $Q(\pi)/Q$ would be a counterexample for this. I can see that there are two automorphisms ...
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23 views

Show that $N(H):=\{g\in G; gHg^{-1}=H\}$ is subgroup of $G$

Let $G$ be a group and $H$ subgroup of $G$, $N(H):=\{g\in G; gHg^{-1}=H\}$ I need to prove that $N(H)$ is subgroup of $G$. It's almost the same question like :$\forall g \in G, gHg^{-1} = H ...
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31 views

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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42 views

Exact functors preserve free modules?

Let $R$ be a principal 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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26 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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56 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
20 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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47 views

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

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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22 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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17 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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82 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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29 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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59 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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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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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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74 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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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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19 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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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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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
20 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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52 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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1answer
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
86 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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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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24 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 $ex=xe=x$, ...
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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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883 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
11 views

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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114 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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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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49 views

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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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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139 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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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 ...
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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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36 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