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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how to show the following is euclidean domain…??

Which of the following is Euclidean domain?? 1.$Z_{2}[i][x]$ 2.$Z_{3}[i][x]$ 3.$Z_{5}[i][x]$ 4.$Z_{15}[i][x]$ And how to we prove this?? I know the 2 properties of Euclidean domain..except that ...
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29 views

Getting the least possible order of a group from existing cyclic subgroups or elements.

Let's say I have finite group $G$. Let it have have following subgroups: $<a> = \{e, a\}$, $<b> = \{e, b\}$, $<c> = \{e, c, c^2\}$, $<d> = \{e, d, d^2, d^3\}$ I can for sure ...
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3answers
432 views

Proving that the Gaussian integers have no zero divisors

The Gaussian integers are the ones of the form $m + ni$ where $m,n$ are both integers. I need to show that given any two Gaussian integers $a$ and $b$, $ab = 0$ must imply that $a = 0$ or $b = 0$. I ...
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1answer
202 views

Why is the naive notion of a product ideal not necessarily additively closed? [duplicate]

Considering the product ideal $IJ = \{ \sum_{i=1}^n a_ib_i | a_i \in I, b_i \in J \forall i\}$, I've always seen it written that the more naive notion $IJ = \{ ij | i \in I, j \in J\}$ is not an ideal ...
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46 views

Ruler and compass constructions and fields

Use the fact that $\alpha=$cos$(2\pi/5)$ satisfies the equation $x^2+x-1=0$ to conclude that the regular $5$-gon is constructible by straightedge and compass. So, the polynomial $x^2+x-1$ is ...
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24 views

How do I know the cyclic subgroups of G generated by a are equal?

I had a question of understanding that I was hoping you guys could help me with. Suppose there exists a group in the integers. For the sake of my understanding, let's say the group is, G = $Z_{60}$. ...
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35 views

Showing that a polynomial over subring is reducible

Suppose that $R_1\subseteq R_2$, and both are integral domains. Further suppose that $R_2$ is a field, where each element $r\in R_2$ is a zero of a polynomial in $R_1[x]$ with the leading coefficient ...
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89 views

showing that $G$ is nilpotent.

suppose $G$ is a finite solvable group,then $G$ is nilpotent if and only if all Hall subgroups of $G$ which its indices are power of a prime number are normal. suppose $G$ is a solvable finite group ...
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1answer
52 views

First Isomorphism Theorem for Rings

I am trying to get a deeper understanding of the First Isomorphism Theorem, but am having trouble seeing the "natural mapping" that my textbook says exists. Upon looking it up more online I came ...
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1answer
58 views

showing that $G$ is not solvable.

suppose $G$ is a finite group and $1\neq a \in G$ ,$1\neq b \in G$ and $O(a)$ ,$O(b)$ ,$O(ab)$ every two of them are relatively prime ,then $G$ is not solvable. my Idea:I suppose $G$ is solvable and ...
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32 views

Verification of an R-Module isomorphism between $R^n$ and its dual

With one step at a time, I am getting slightly more used to $R$-Modules. Let $R$ denote a commutative Ring with $\mathbb{1}$ and $n$ a natural number. For the tuple $a:= (a_i)_{i=1}^n \in R^n$ we ...
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17 views

Show $X_1^5+X_1^2X_2+X_1X_2+X_2$ is irreducible

For a field $K$, is the polynomial $X_1^5+X_1^2X_2+X_1X_2+X_2$ irreducible in $K[X_1,X_2]$? I think I have to show it for $K(X_1)[X_2]$ but I don't know how to do that.
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76 views

$X^3+2$ is irreducible in $\mathbb{F}_7[X]$

Let $I=(X^3+2)$ be the principal ideal of $\mathbb{F}_7[X]$ generated by $X^3+2$. Show that $X^3+2$ is irreducible in $\mathbb{F}_7[X]$. Can someone give me the first step on how to do this ...
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2answers
119 views

Left ideals of matrix rings are direct sum of column spaces?

Let $\mathbb K$ be a field and $M_n(\mathbb K)$ be the ring of the $n\times n$ matrices with entries in $\mathbb K$. Let $C_j\subset M_n(\mathbb K)$ be the subspace of all matrices which have all ...
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83 views

Module and group ring: definitions and notations

I apologize in advance for the stupid questions and the bad English, but I've started studying math few months ago. I've some problems with the definitions of group ring, modules and their notations. ...
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96 views

Why every central simple algebra has a splitting field

For central simple $F$-algebra $A$, where $F$ is a field, a field $E$ such that $F \subset E$ is called a splitting field if $A \otimes E$ is isomoprhic to $M_n (E)$ for some $n$. Why is algebraic ...
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42 views

Prove $\bigcup _{k=1}^n I_k$ is in ideal

Let $n$ ideals such that $I_1 \subset I_2 \subset \cdots \subset I_n$. Prove that $J=\bigcup _{k=1}^n I_k$ is also an ideal. We need to show three things: $0\in J$. Trivial... $a,b \in J \implies ...
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75 views

On prime ideals in semigroups

I have a question about the following exercise: Suppose $I$ is an ideal (two-sided) in a semigroup $S$. Prove that $I$ is prime (which means that from $aSb\subset I$ for some $a,b\in S$ it follows ...
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Show that no linear polynomial divides $x^k + x^{k-1} + \cdots + 1$ with $k\ge 2$ even

Let $f(x) = x^k + x^{k-1} + \cdots+ 1 \in \mathbb{Q}[x]$, $k\ge 2$ and even. Show there's no linear polynomial which divides $f(x)$. A start: Lets assume by contradiction there's a linear ...
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26 views

A subgroup of multiplicative group of a field

I can't seem to find a solution to this: Let $ K $ be a field of characteristic not equal to $ 2 $, $ K^* $ be its multiplicative group and $ K^{*2} = \{k^2, ~ k\in K^*\}$ Prove that if $ a \notin ...
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49 views

Is a projective $R[G]$-module a projective $R[H]$-module if $H$ is a subgroup of $G$?

I have a ring $R$ of characteristic $0$ and a finite group $G$. Let $H$ be a subgroup of $G$. Question: If $M$ is a projective $R[G]$-module where $R[G]$ is the usual group ring then is $M$ ...
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what is the set $\mathbb R[X]$ defined as?

Can someone quickly tell me what the set $\mathbb R[X]$ is defined as, where $\mathbb R$ is the set of real numbers? Is it $$\mathbb R[X]=\{ a_nX^n+.........+a_1X+a_0\mid a_n,...,a_0 \in\mathbb ...
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1answer
23 views

Representation of an inverse

I'm facing the following problem: Let $ K $ be a field of characteristic not equal to $ 2 $. Prove that if $ \alpha \in K $ is representable as $ x^2 - ay^2 $, then so is $ \alpha^{-1} $ Well, I ...
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23 views

Show that $a$ is irreducible iff $au$ is irreducible where $u$ is invertible

$R$ is integral domain. Show that $a$ is irreducible iff $au$ is irreducible where $u\in R^*$. My Try: Lets assume $a$ is irreducible and $au = bc \implies a=bcu^{-1}$. We know that $u\in R^*$ so ...
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39 views

Prime ideals in $\mathbb{Q}[X]$

Could you tell me why prime ideals in $\mathbb{Q}[X]$ are of the form $(q(x))$ where $q \in \mathbb{Q}[X]$ is irreducible?
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33 views

$R$ integral domain : $u\in R^*, a \text{ is prime} \iff au \text{ is prime}$

$R$ integral domain : $u\in R^*,\; a \text{ is prime} \iff au \text{ is prime}$ I started by looking at $auu^{-1}$. What should I do next? I'd be glad for help. Note: $u \in R^*$ meaning is $u$ ...
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178 views

Any group which is of prime order is a cyclic group

I don't know how to prove this: Any group which is of prime order is a cyclic group. What fact should I use to prove this?
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If N and every subgroup of N is normal in G then G/N is abelian .

Let $N$ be a normal subgroup of $G$ such that every subgroup of $N$ is normal in $G$ and $C_G(N)\subseteq N $ .Prove that $G/N$ is abelian. I think we need to use that every subgroup of $N$ is ...
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60 views

Proving that an ideal is prime - is it correct?

I need to prove that although $X^2 + 3X +1 \in \mathbb{Z} [X]$ is irreducible, the ideals $(5,X^2 + 3X +1 )$ and $(11, X^2 + 3X +1)$ are not prime. I know that an ideal $I$ is prime iff ...
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60 views

$R/\mathfrak p$ not always a UFD [closed]

I am looking for a nice counterexample that for a UFD $R$ and $\mathfrak p\subset R$ a prime ideal, $R/\mathfrak p$ is not always a UFD as well.
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92 views

$\mathbb{F}_{2}$ is the subgroup of $\mathbb{F}_{3}$

Fix $r\in \mathbb{N}$ and let $\mathbb{F}_{r}=\langle g_{1}, ...,g_{r}\rangle$ be the rank-r free group. Then, How to prove $\mathbb{F}_{2}$ is the subgroup of $\mathbb{F}_{3}$? (Or maybe you can ...
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41 views

does $\phi(-1) = -1$?

I want to show that $\mathbb{R}$ and $\mathbb{C}$ are not isomorphic when considered groups under multiplication. My idea was to show that there is no real number that square to $-1$. Suppose an ...
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60 views

Is this a subgroup?

For a fixed element $a$ of a group $G$, prove or disprove: The set $H = \{xa\mid x \in G\} $is a subgroup of $G$. Clearly it is nonempty since if $e$ is the identity of $G$, then $ea = a$. Now ...
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106 views

proof of a useful counting result in group theory

Let G be a finite group, H a subgroup of G satisfying |G| |̸| [G : H]!. Prove there exists a normal subgroup N of G satisfying 1 < N ⊂ H. maybe the General Cayley's Theorem works. I am not sure. ...
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236 views

Let R be a ring with unity and S be the set of all units in R

I need to prove two things (a) prove or disprove that $S$ is a subring of $R$ (b) prove or disprove that $S$ is a group with respect to multiplication in $R$ For (a) (1) it is clearly that $S$ is ...
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210 views

Prove that the set of all functions is not a group under function composition.

Consider the set $F$ of all functions from $\{1,2,3\}$ to $\{1,2,3\}$. There are $3^3= 27$ of them. Prove this set is not a group under function composition. I thought that it violates the ...
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40 views

If $\tau_1=(a\space b), \tau_2=(c\space d)$, why is $\tau_1\tau_2=(d\space a\space c\space )(a\space b\space d)?$

For two permutations $\tau_1=(a\space b), \tau_2=(c\space d)$, why is $\tau_1\tau_2=(d\space a\space c\space )(a\space b\space d)? \text{ (where }a,b,c,d \text{ are all distinct)}.$ I'm fairly new ...
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689 views

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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Factoring in $\mathbb{Z}[\sqrt{2}]$

How would one factor a number, say $9+4\sqrt{2}$ in $\mathbb{Z}[\sqrt{2}]$? This is what I've attemped to do: $$(a_1+b_1\sqrt{2})(a_2+b_2\sqrt{2}) $$ $$a_1a_2+a_1b_2\sqrt{2}+a_2b_1\sqrt{2}+2b_1b_2$$ ...
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119 views

How to prove that the center of a group is not a maximal subgroup?

How to prove $Z(G)$ is not a maximal subgroup of $G$? So I used the fact that $Z(G)$ is a subset of the $C(a)$ which is a subgroup of some group $G$. Is that sufficient? If not can you provide a ...
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348 views

Is this study plan sufficiently general, or overly specialized? [closed]

My current study plan is in order below. I will be completing these textbooks in this order one at a time. I have been told that I don't have textbooks in my plan that approach topology in a general ...
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50 views

How to show that this is a group isomorphism?

I want to show that if for some $n \in \mathbb{Z}$, the map $f_n\colon (\mathbb{Z},+) \rightarrow (\mathbb{Z},+), x \mapsto nx$ is a group isomorphism, then $n \in \{-1,1\}$, without using anything ...
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Epimorphisms detect structure-preserving maps

Suppose that $U: \mathsf{Top} \to \mathsf{Set}$ is the forgetful functor. I believe that for a topological quotient map $\pi:R \to S$, and a map $\phi:U(S)\to U(T)$, we have that $\phi \circ U(\pi)$ ...
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363 views

Presentation of group equal to trivial group

Problem: Show that the group given by the presentation $$\langle x,y,z \mid xyx^{-1}y^{-2}\, , \, yzy^{-1}z^{-2}\, , \, zxz^{-1}x^{-2} \rangle $$ is equivalent to the trivial group. I have tried ...
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199 views

The Normalizer and Centralizer of Groups

recently my teacher has introduced us two new Theorems called the N/C Theorem and the Normalizer Theorem. N/C states: Let $H<G$ and $N(H)$ be a normalizer of H in G. Let $C(H)$ be the centralizer ...
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1answer
79 views

Characterization of free modules

Let $M$ be a finitely generated module over a commutative ring $A$. Is it true that if there exists a positive integer $n$ and a pair of homomorphisms $\pi:A^n\rightarrow M$ and $\phi:A^n\rightarrow ...
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23 views

Z(G)<C(a)<=G why is this true?

Can anybody provide a proof of this? Also is it true that for an arbitrary y in G, but not in Z(G) that we obtain Z(G)< C(a) < G? (Note: C(a) is the centralizer and Z(G) is the center).
2
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1answer
65 views

Show that it is an integral domain

I want to show that $$\mathbb{C}[X,Y,Z]/(Y-X^2,Z-X^3) \text{ is an integral domain }.$$ How can I do this? Do I have to find a homomorphism from $\mathbb{C}[X,Y,Z]/(Y-X^2,Z-X^3)$ to an integral ...
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69 views

Problems with proving P. Hall theorem for finitely presented groups

I've problems with proving P. Hall theorem about finitely presented groups. I would like to find a proof different from the one written in Robinson textbook. I can't prove this theorem and in ...
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42 views

Group homomorphisms and vector space homomorphisms

I have a certain abelian group G and a certain finite field and I have shown that G is a vector space over this field. Now I'm asked to show that every group homomorphism is a vector space ...