Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, and other advanced topics.

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Derived subgroup of a finite non-Abelian p-group is proper?

How do I show that the derived subgroup of a finite p-group is always proper? In Abelian groups, it's trivial. In non-Abelian groups, my intuition is that there should be some way to relate G/Z(G) to ...
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25 views

Steps to construct the Field of fractions of Gaussian Integers $\mathbb{Z}[i]$

i don't know how to construct such field $\mathbb{Q[i]} $ from $\mathbb{Z[i]}$. I know the following: $(a+bi,c+di)\sim (m+ni,r+si)$ iff $(a+bi)(r+si)=(c+di)(m+ni)$ is the equivalence relation and if ...
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21 views

Existence of projectives in the category of torsion abelian groups

Consider the category of torsion abelian groups. This category doesn't have enough projectives by the following argument. Suppose $C_2$ (cyclic group of order 2) is the homomorphic image of a ...
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1answer
23 views

Quotient Ring and finite fields

How is a quotient ring $\mathbb Z/p^e\mathbb Z$ (where p is prime and $e>2$) different from a finite field $\mathbb F_{p^e}$? When they are both rings, have the same elements? I thought a finite ...
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27 views

Show that an extension is separable

Let $K$ be a field with $\operatorname{char} K=p$, where $p$ is a prime, and let the degree of the extension $K \leq L$ be coprime to $p$. How can I show that the extension is separable?? Could you ...
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31 views

Atiyah & MacDonald on local Noetherian and Artinian rings - sanity check.

In the chapter on Artinian rings in "Introduction to Commutative Algebra" by Atiyah and MacDonald, we have: Proposition 8.6. Let $(A,\mathfrak{m})$ be a local Noetherian ring. Then exactly one of the ...
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1answer
21 views

Correspondence between ideals of $R$ and $D^{-1}R$

Let $R$ be an integral domain, and $D\subset R$ be a multiplicatively closed subset such that $1\in D$ and $0\not\in D$ . Prove/disprove that there is a one-to-one correspondence between the ideals of ...
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28 views

A subgroup of $\mathbb{Z}$

Let $A$ be a subgroup of $\mathbb{Z}$. Show that $A=\{0\}$ or $A\cong\mathbb{Z}$. My intuition is to do something with the generator(s) of A (maybe it's not the best thought), but I have no idea how ...
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20 views

Trouble on Aluffi's Exercise 5.5, Chapter 6: direct summand over a local ring is a free module

The overall goal of the problem is the following: Let $R$ be a local commutative ring (so that there is a unique maximal ideal $\mathfrak{m}$) and let $M$ be a finitely generated $R$-module. Assume ...
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16 views

Prove that $\Bbb F_p^\times$ is equal to Miller–Rabin primality test for prime number

I want to prove, that $\Bbb F_p^\times = MRP(p)$. I think, that I have to start with this statement: $\{a \in \Bbb F_p^\times | a^2 = 1 \} = \{1; -1\}$ But I do not know how to continue this idea.
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Suppose that $R$ is a commutative ring with unity, $a \in R$, and $\varphi(r) = ar$ defines a ring homomorphism. Prove that $a$ is idempotent.

Suppose that $R$ is a commutative ring with unity, $a \in R$, and $\varphi(r) = ar$ defines a ring homomorphism $\varphi$. Prove that $a$ is idempotent, i.e. that $a = a^{2}$. This is exercise 15 ...
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1answer
18 views

If $\lvert g \rvert=m$ is finite then prove that $ng=0$ if and only if $m\mid n$.

Let $G$ be an abelian group and let $g \in G$. If $\lvert g \rvert=m$ is finite then prove that, for $n\in \mathbb Z$, $ng=0$ if and only if $m\mid n$. I think this amounts to proving that: $$ng=0 ...
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33 views

Complex coefficients in a quadratic equation [on hold]

Can the general solution to the quadratic equation be used if the coefficients are complex?
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37 views

Polynomials over $\mathbb{F}_2$ without multiplicity 1 factors

Let $f \in \mathbb{F}_2[T]$ such that both $f$ and $f + 1$ have the property that every irreducible factor in the unique factorization domain $\mathbb{F}_2[T]$ appears with multiplicity at least $2$. ...
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1answer
17 views

Calculation of the Cosets of the Kernel

Let $g:\Bbb{Z}_3 \times\Bbb{Z}_4 \to \Bbb{Z}_3$ where $g((a,b)) =a$ . What are the cosets of the kernel? I understand that a kernel is the set of elements that map to the identity element. Any ...
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2answers
29 views

Difference between Integral Domains and Fields.

Can someone please help me in figuring out how all fields are integral domains but not all ID are fields? My course assumes IDs to be commutative with unity but fields require all elements to have a ...
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Let $F$ and $K$ be fields. If $F\subseteq K$ and $r\in K$ s.t. $r^2$ is algebraic over $F$. Then $r$ is algebraic over $F$.

Let $F$ and $K$ be fields. If $F\subseteq K$ and $r\in K$ s.t. $r^2$ is algebraic over $F$. Then $r$ is algebraic over $F$. Assume polynomial $p(x)\in F[x]$ s.t. $p(r^2)=0$ If $r\in K$ and $r^2$ is ...
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20 views

Finding the kernel of $\phi:\Bbb{Z}_6\to\Bbb{Z}_2 $ given by $\phi(x)=$ the remainder of $x$ when divided by $2$

Let $\phi:\Bbb{Z}_6\to\Bbb{Z}_2 $ be given by $\phi(x)=$ the remainder of $x$ when divided by $2$. I have become fairly confident with calculating the left and right cosets, but what is the kernel ...
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1answer
31 views

What is the name of this terminology?

Let $G$ be the group generated by a set $X=\{x_1,\cdots,x_n\}$. Then each element can be (not necessarily uniquely) written as a product of the form $x_{j_1}^{e_1}\cdots x_{j_k}^{e_k}$, where each ...
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24 views

How to show that an ideal of $F[x]$ containing an irreducible polynomial of degree $n$ and a nonzero polynomial of degree $<n$ is $F[x]$?

Let $F$ be a field and suppose that $I$ is an ideal of $F[x]$ which contains an irreducible polynomial of degree $n$ and a nonzero polynomial of degree less than $n$. Show that $I=F[x]$. I can't ...
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371 views

Each element is a square of some element

I have to show that each element of $\mathbb{F}_{2^n}$ is a square of some element. Could you give me some hints how I could do that??
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25 views

Acyclic chain complex and contracting chain homotopy

Let $R$ be a Ring and $(C_k, d_k)_{k\geq0}$ a acyclic chain complex of free modules, meaning $im(d_{k+1})=\ker(d_k)$ for all $k$. I want to show that there is a family of R-module-homomorphisms ...
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61 views

Suppose that $[G:H]$ is a prime integer, and that $g \notin H$. Prove that H is normal in G.

Let H be a subgroup of a group G. Let $k,g \in G$ such that $gH = Hk$. Suppose further that $[G:H]$ is a prime integer, and that $g \notin H$. Prove that H is normal in G. I have totally no idea at ...
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What is the 'Hom-description'?

I am trying to learn about the 'Hom-description' of the class group $Cl(A)$ of an $R_K$-order $A$ in $K[G]$ where $K$ is a number field with ring of integers $R_K$ and $G$ is a finite group. I've ...
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17 views

Question about permutation.

Suppose a and b are permutations of the same cycle type. Why aligning them on top of one another and interpret it as a two line representation of permutation gives me a permutation that will conjugate ...
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38 views

If $p$ and $q$ are prime numbers and $m\gt n$ show that $\sqrt[m]{p}\notin \mathbb Q(\sqrt[n]{q})$

If $p$ and $q$ are prime numbers and $m\gt n$ show that $\sqrt[m]{p}\notin \mathbb Q(\sqrt[n]{q})$ I really have no idea how to prove this problem. I started to consider: Assume $\sqrt[m]{p}\in ...
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If $V$ is a $\mathbb CG$-module then we may take $\rho(g)$ as a diagonal matrix?

If $G$ is a group and $\mathbb K$ is a field let $\mathbb KG$ be the usual group ring. We know a representation $\rho:G\longrightarrow GL(V)$, where $V$ is a $\mathbb K$-vetor space, is the same as a ...
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1answer
66 views

Is $\mathbb{C}\bigotimes_\mathbb{R}\mathbb{C}\simeq \mathbb{C}\bigotimes_{\mathbb{C}}\mathbb{C}$? [duplicate]

I'm trying to see if for several cases changing the ring in a tensor product affects the result or doesn't. Now I'm trying to prove $\mathbb{C}\bigotimes_\mathbb{R}\mathbb{C}\simeq ...
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1answer
23 views

Suppose that $N_1$ is a normal subgroup of $G_1$. Is the image $f(N_1)$ of $N_1$ a normal subgroup of $G_2$? [duplicate]

Let $f:G_1 \to G_2$ be a homomorphism between multiplicative groups. Suppose that $N_1$ is a normal subgroup of $G_1$. Is the image $f(N_1)$ of $N_1$ a normal subgroup of $G_2$?
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Showing that an ideal is principal.

I need to show that the ideal $(3 + i , 6)$ is principal in $\mathbb{Z}[i]$ and find its generator. So I know that I need to find an element such that $<t> = (3 + i , 6)$. My intuition tells me ...
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1answer
51 views

tensor product and commutation, category theoretical argument

It is a well-known fact that taking direct limits commutes with tensor products in the following sense: Let $I$ be a directed set and suppose for every $i\in I$ we have modules $M_i,N_i$ over a ring ...
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Properties of an infinite group with an infinite cyclic normal subgroup

Let $G$ be an infinite group with an infinite cyclic normal subgroup $H$ such that $|G/H|=2$ and is cyclic. Show that $G$ is isomorphic to one of $\mathbb{Z},\mathbb{Z}\times\mathbb{Z_2},D_\infty$. ...
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Subfield Criteria - Proof or Counterexample

I am interested in whether the following claim is true for all fields $F$: Conjecture: A subset $X\subset F$ is a subfield if and only if (1) $1\in X$, (2) $x,y\in X\Rightarrow x-y\in X$; and (3) ...
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1answer
23 views

Proving that this mapping is one to one

Let $Q$ be the field of quotients of the Gaussian integers (integer complex numbers) and let $R$ be the the set of all complex numbers of the form $a +bi$ such that both $a,b$ are rationals I have to ...
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1answer
29 views

Isomorphism with Euler phi function

Let $m_i > 1$, where $1 ≤ i ≤ n$, be integers, pairwise relatively prime. Let $m = m_1 \cdots m_n$. Let $\phi(m)$ denote the order of the group $(Z/mZ)^×$. The function $\phi : Z_+ → Z_+$ is called ...
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1answer
28 views

An algebraic curve $C$ as a cover of $\mathbb{P}^1$ over $\bar{\mathbb{Q}}$

Could someone explain what it means for an algebraic curve $C$ to be a cover of $\mathbb{P}^1$ over $\bar{\mathbb{Q}}$, ramified over $n$ points?
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Concerning Ideals and invertible elements in a commutative ring

Here is the problem that I have: Let $R$ be a commutative ring with unity and let $I$ be an ideal in $R$. Prove that $I=R$ if and only if $I$ contains some invertible element of the ring $R$. Here ...
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1answer
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How to prove a subgroup is normal?

Prove that $D$ is a normal subgroup of $C$ if $C=S_3 \times \Bbb Z_4$ and $D=\langle((132),2)\rangle$. I know to prove a subgroup is normal you have to show aH=Ha, but I'm just not sure how to do this ...
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If $f$ is a unit in a polynomial ring then $a_0$ is unit and all other coeficients are nilpotent. [duplicate]

I'm trying to prove the converse of the following theorem. I think suggestion available at this website are mistaken or I didn't understand them correctly. Theorem. Let $R$ be a commutative ring with ...
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1answer
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$H, K, HK$ is a subgroup of G, then $|G:H\cap K|$ divides $|G:H||G:K|$.

I came across this fact that says if $H, K, HK$ is a subgroup of G, then $|G:H\cap K|$ divides $|G:H||G:K|$. I can prove that $|G:H\cap K| \leq |G:H||G:K|$. However, I can't get the divisor ...
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32 views

Quotient field of gaussian Integers

Let $D$ be the set of all gaussian integers in the from of $m+ni$ where $m,n \in Z$ Carry out the construction of the quotient field $Q$ for this integral domain.Show that this quotient field is ...
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1answer
35 views

Extensions of $\mathbb{Z}/(m)$ by $\mathbb{Z}$

I know that $\text{Ext}_{\mathbb{Z}}^1(\mathbb{Z}/(n), \mathbb{Z}) \cong \mathbb{Z}/(n)$. I am trying to use this to show that the extensions of $\mathbb{Z}/(n)$ by $\mathbb{Z}$ are $$0 \to ...
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Proof or counterexample: If $F\subseteq K$ and $r\in K$. If $[F(r):F]=4$ then $F(r)=F(r^3)$.

$F$ and $K$ are fields. Proof or counterexample: If $F\subseteq K$ and $r\in K$. If $[F(r):F]=4$ then $F(r)=F(r^3)$. I think I need to find a polynomial in $F(r^3)[x]$ that has $r$ as a root. I ...
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1answer
26 views

Is binary isomorphic to decimal representation?

My friend and I were just talking about whether decimal representation isomorphic to binary one. He said that it is true since there is a obvious 1-1 relationship between them. But how about ...
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what is the basic difference between a mapping and a function?

what is the basic difference between a mapping and a function? many say they are same but the opposite views are also seen. is mapping a restricted version of a function?
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1answer
23 views

Questions about ring of smooth functions

First of all, this is a homework problem. Let $C^{\infty}(\mathbb{R})$ denote the ring of smooth functions. Let $I_n$ denote the set of $f\in C^{\infty}(\mathbb{R})$ such that $$f^{(k)}(0)=0, \ 0 ...
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1answer
65 views

Why $F[x]/p(x)$ would contain $F$?

I am reading Abstract Algebra by Hungerford, and I am really confused about how we can extend a ring to a bigger ring. Here's what I got from the book: $F$ be a field and $p(x)$ be a nonconstant ...
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2answers
38 views

Proving the kernel of a homomorphism

I am trying to show that $\ker(\varphi)=<2\pi>$ where $\varphi: \mathbb{R} \to \text{SL}_2(\mathbb{R})$ by $\varphi(x)=\begin{bmatrix} \cos(x) & \sin(x) \\ -\sin(x) & \cos(x) ...
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0answers
36 views

Cyclotomic Extensions and Minimal Polynomials

Is the following an isomorphism: $ \mathbb {Q}(\zeta_m)/ \mathbb{Q} \cong \mathbb{Z}[X]/ X^m -1 $ Where $ \mathbb {Q}(\zeta_m)/ \mathbb{Q}$ is a field extension created by adjoining $\zeta_m$, an ...
0
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2answers
29 views

Matrix rings and ideals

How would one go about checking if a given 2 x 2 matrix is an ideal. I am unclear as to what an ideal is and would like to know the steps in order to make the verification. Also, if it helps, I had ...