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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Buchberger algorithm and ideals

I'm working on Groebner basis using the book Ideals, Varieties and Algorithms. I'm interested in this problem : Let $\mathbb{Q}[x,y,z]$ with the graded lexicographic order with $x>y>z$ For ...
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$\mathbb{Z}[i]\otimes_{\mathbb{Z}}\mathbb{R}$ is isomorphic to the complex numbers

I am new to tensor poducts (of modules over a commutative ring with identity) and need to understand the following example to continue with the actual exercises in my material. Namely, I need to know ...
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Two normal subgroups and isomorphism theorem

Question Let $N_1$ and $N_2$ be normal subgroups of $G$. Prove that $N_1N_2/(N_1\cap N_2) \cong (N_1N_2/N_1)\oplus (N_1N_2/N_2)$. I think the homomorphism must be $\phi : N_1N_2 \to ...
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Group of order $p^2$

Let $G$ be a group of order $p^2$ where $p$ is a prime. I want to show that either $G$ is cyclic or is the product of two cyclic groups of order $p$. After some work, the problem reduces to showing ...
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$\mathbb{R}[x]/\langle x^2+1\rangle$ and $\mathbb{R}[x]/\langle x^3 +x+1\rangle$ are isomorphic?

$\mathbb{R}[x]/\langle x^2+1\rangle$ and $\mathbb{R}[x]/\langle x^2 +x+1\rangle$ are isomorphic or not? I guess these are isomorphic as they are isomorphic to the field of complex number. But how can ...
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A slight confusion in Galois Theory

Let $K=\mathbb{Q}$ and consider a cyclic extension $L$.(For example say, the splitting field of the polynomial say $f=x^3+x^2-2x-1$). Now consider a cyclotomic extension of $\Phi_3=X^2+X+1$. Let us ...
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Are the following two ideals equal? How to prove it, or show they are not?

$I= \langle x-y^2, x-y^3, x-y^4,... \rangle, $ and $J=\langle x-y^2, x-y^3\rangle$. Obviously $J \subset I$, but what about the reverse inclusion?
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irreducibility over integral domains

A polynomial $f(x)=g(x).h(x)$ over $D$ ,where $g(x)$ or $h(x)$ must be a unit in $D[x]$ and $D$ is an integral domain, then we say that $f(x)$ is irreducible polynomial over $D$. Since $\mathbb Q$ is ...
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Line bundle trivial on fibers then already trivial.

$\require{AMScd}$ Im currently reading Milne's notes about Abelian varieties. On page 26 he proves the following theorem: Let $V$ and $T$ be varieties over $k$ with $V$ complete, and let ...
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Localization of triangulated categories

I have been reading from the Stacks project, and Lemma 13.5.4. says: Let $F : \mathcal{D} \to \mathcal{D}'$ be an exact functor of pre-triangulated categories. Let $$ S = \{f \in ...
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Product group isomorphism

Let $H$ and $K$ be subgroups of a group $G$, and let $f:H\times K \to G$ be the multiplication map, defined by $f(h,k)=hk$. Show that $f$ is an isomorphism from the product group $H \times K$ to $G$ ...
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7 views

Equivalent definitions of primary ideals [duplicate]

In Atiyah-Macdonald, an ideal $I\in R$ is primary if $fg\in I$ implies that $f\in I$ or $g^n \in I$ for some $n$. The ideal $I$ is $P$-primary for a prime ideal $P$ if it is primary and $\sqrt{I}=P$. ...
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28 views

Isomorphisms between endomorphism algebras

Assume that $R$ and $S$ are associative $\mathbb{C}$-algebras with unit $1_R$ and $1_S$, respectively. In addition, assume that $_RM$ is a simple left $R$-module and $_SN$ is a simple left $S$-module. ...
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What is $\mathbb{R}/\mathbb{Z}$ going to be?

Still kind of kind of not getting the hang of this. When we see $\mathbb{Z}/\mathbb{2Z}$ for instance, I can relate it to the set of integers modulo $2$. I look at $\mathbb{R}/\mathbb{Z}$ and I am ...
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38 views

Are they isomorphic to each other?

Let $R$ a ring. Suppose that $\phi : R[x]\rightarrow R$ is an epimorphism. Does it follow then that $R[x]/\ker\phi \cong R$ ? Or does it hold the same as for the groups?
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How many Isomorphisms are there from $\mathbb{Z}_3\oplus\mathbb{Z}_5$ to $\mathbb{Z}_{15}$

How many Isomorphisms are there from $\mathbb{Z}_3\oplus\mathbb{Z}_5$ to $\mathbb{Z}_{15}$ We have generator of $\mathbb{Z}_3\oplus\mathbb{Z}_{15}$ is $1$ and there $\phi(15)=8$ generators of ...
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22 views

A group-ring is commutative if and only if that group is abelian

Problem says: Let $R$ be a commmutative ring and $G$ a group. Prove that $R[G]$ is commutative if and only if $G$ is abelian. I solved ($\Rightarrow $) direction as follow: Suppose that ...
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10 views

$[k(\alpha):k]=p, [k(\beta):k]=q$, $p>q$ are primes, then $k(\alpha,\beta)=k(\alpha+\beta)$

Let $p>q$ be primes. Suppose $L\mid_{k}$ is an algebraic extension and $\alpha,\beta\in L$ are such that $[k(\alpha):k]=p$, $[k(\beta):k]=q$ and characteristic of $k$ is coprime with $p$. Show that ...
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32 views

In a loop, if $(xy)x^r = y$ then $x(yx^r)=y$

Consider a loop $L$, that is, a quasigroup with an identity, and recall that a quasigroup $L$ is a set together with a binary operation such that, for every $a$ and $b$ in $L$, the equations $ax=b$ ...
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18 views

Are there any integral domains in which irreducible elements are easily identified?

In every integral domain I've studied so far irreducible elements have been impossible to quickly identify in general with any known procedure. Is there an integral domain for which such a procedure ...
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31 views

$R_{a} = R[x]/(x)$ isomorphic to $R_{b} = R[x]/(x-1)$

I am looking at the following two rings: $R_{a} = R[x]/(x)$ and $R_{b} = R[x]/(x-1)$. I was told that these two rings were isomorphic, but I don't see why. Is this due to the minimal polynomials? ...
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39 views

Ideal of a product ring?

I am trying to prove whether or not the ideal generated by $\langle (2,2)\rangle$ is a prime ideal of $\mathbb Z_4\times \mathbb Z_4$? My issue is I'm not sure how to do the coordinate ...
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Endowing module with algebra

Given a module $M$ over a ring $R$, is it possible to endow $M$ with an operation $M^{2} \to M$ that turns $M$ to an algebra that is not the trivial $m n \equiv \mathbf{0}$ identically? So far I have ...
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39 views

Prove $Q[x]/(x^2+4)$ is isomorphic to $Q[x]/(x^2+1)$

I've been asked to prove Q[x]/(x^2+4) is isomorphic to Q[x]/(x^2+1); I've looked at lots of similar solutions, but haven't been able to understand this. I know each ring is the quotient ring for their ...
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23 views

Characteristic of a product ring?

Let A and B be commutative rings with unity where char(A)=n and Char(B)=m s.t. n,m ∈ ℤ (and n,m ≠ 0). Prove or give counter example: if k ∈ ℤ+ and n,m both divide k, then Char (A x B)| k. Here was ...
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24 views

Counting permutations in $S_n$ with $1,2,..,k$ all in same cycle

The number of permutations in $S_n$ for which the first $k$ items $1,2,...,k$ are all in the same cycle can be shown (by a somewhat tedious argument) to be $n!/k.$ I'm looking for less computational ...
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43 views

There are at most finitely many square-free integers $d\not\equiv 1\pmod{4}$ such that $\mathbb{Q}(\sqrt{d})$ is a Euclidean field

My book's exercise is about proving that there are at most finitely many square-free integers $d\not\equiv 1\pmod{4}$ such that $\mathbb{Q}(\sqrt{d})$ is a Euclidean field (with respect to the norm). ...
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24 views

abstract algebra: finite fields and galois group [on hold]

I'm trying to find the order and all generators of the Galois group of $GF(256)$ over $GF(16)$? How to approach this problem?
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33 views

If $K=F(K^p)$ is a finite extension and $\{a_1,\ldots,a_n\} \subset K$ linearly independent then so is $\{{a_1}^p,\ldots,{a_n}^p \}$

Suppose that $F$ is a field of characteristic $p$. Let $K/F$ be a finite extension and $K=F(K^p)$, where $K^p:= \{x^p\mid x\in K\}$. Suppose $\{a_1,\ldots,a_n\} \subset K$ is linearly independent ...
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61 views

Galois extension of degree $2^n$ [on hold]

Let $k$ be a field and $L=k(\sqrt{a_1},\sqrt{a_2},\cdots,\sqrt{a_n})$ with $a_i\in k$. Suppose $[L:k]=2^n$. Let $a=\sqrt{a_1}+\sqrt{a_2}+\cdots\sqrt{a_n}$. Show that $L=k(a)$.
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Norm of ideal belongs to the ideal [duplicate]

Suppose that $D$ is any number ring (i.e. $D=\mathbb Q(\alpha), \alpha \in \mathbb C$). Let $I$ be any ideal of $D$. Show that $N(I)=|D/I|$ belongs to $I$. How to start? is there a specific fact will ...
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31 views

Intersection of two principal ideals is an ideal and lowest common multiple (if it is a PI)

I think the first part of the proof would go like this: any element in $(a) \cap (b)$ can be written as $ar_1 = br_2$, so multiplying by an element $r \in R$ yields $ar_1r\in aR$ or $br_2r \in bR$, so ...
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Group of translations is a normal subgroup of group $E(2)$ of isometries of $\mathbb{R}^2$?

I'm trying to see that: The group of translations $T=\{t(x)=x+a : a \in \mathbb{R}^2 \}$ is a normal subgroup of group $E(2)$ of isometries of $\mathbb{R}^2$. I know the definitions of a normal ...
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1answer
27 views

Tensor Product of $\mathbb{Q}[\sqrt{2}]$.

How can one show that $\mathbb{Q}[\sqrt{2}] \otimes_{\mathbb{Q}[\sqrt{2}]} \mathbb{Q}[\sqrt{2}] \simeq \mathbb{Q}[\sqrt{2}]$ (which is a $2$ dimension vector space over $\mathbb{Q}$) and ...
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9 views

Intersection projective subspaces [on hold]

Im struggling with understanding projective spaces, I know there is a connection between projective and vectorial spaces, could someone help me showing that the intersection of projective subspaces is ...
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1answer
17 views

What's group $E(2)$ of isometries of $\mathbb{R}^2$?

I'm trying to prove normal subgroups of the group $E(2)$, but I haven't been given, what the group $E(2)$ of isometries of $\mathbb{R}^2$ is like. What is it like?
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36 views

Proof of the fixed point theorem in group action.

$\textbf{Fixed point theorem:}$ Let $G$ be a $p$-group and let $S$ be a finite set on which $G$ operates. If the order of $S$ is not divisible by $p$, there is a fixed point for the operation of $G$ ...
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A commutative unital ring is a field iff its only ideals are $0$ and $R$

A commutative ring $R$ with unity is a field if and only if its ideals are $0$ and $R$. How can I prove it?
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Group of polynomial $x^4+2$ in $\mathbb Q[x]$

Describe the Galois group of the polynomial $x^4+2 \in \mathbb Q[x]$. I've been able to see how to do this for $x^4-2$ and $x^4+1$ but am unsure how to do this for the polynomial above. Based on the ...
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25 views

How to prove $t(x)-x\in F_{p}$ when $p(x) \in K$, $K$ a field whose characteristic is $p>0$

Suppose $K$ is a field whose characteristic is $p>0$, and $L$ is a finite Galois extension of $K$. Also, we have an endormorphism $h$ such that $h(x)=x^p-x$ for any $x \in L$. Question: If ...
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Classify groups of order 100 [on hold]

So I am currently trying to Classify all groups of order 100 through an extensive proof; and this is as far as I have gotten so far, wondering how to go beyond the fact that both squares (Z4 & ...
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31 views

$F$ is a field, $K$ is the splitting field of all the polynomials over $F$, how to prove that $K$ is an algebraic closure of $F$?

$F$ is a field, $K$ is the splitting field of all the polynomials over $F$, how to prove that $K$ is an algebraic closure of $F$. I know this result, but I don't know the details of the proof, which ...
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Generators of two groups with prime order $p$ already induce all the generators of the product group $G \times H$

Let $G = \langle g \rangle, H = \langle h \rangle$ be two cyclic groups (with $g \in G, h \in H$), both of them of order $p \in \mathbb{N}$, where $p$ is a prime number. I now want to show that ...
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1answer
16 views

Factoring a homomorphism through the quotient by a normal subgroup contained in the kernel

Suppose $f\colon G\to G^{\prime}$ is a group homomorphism. Let us denote the groups additively. It is well know that such a homomorphism always can be 'factored through' the quotient ...
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15 views

Group ring generated by Z and the quarternion group

I want to calculate general nth power of i+j in the group ring. My idea was to find some patterns after calculating some powers of i+j, conjecture the general form of nth power of it and prove it by ...
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40 views

Some examples of "Clean topological spaces'

What is an example of a Hausdorff topological space $X$, not a singleton, such that the ring $C(X)$ of all real (or complex) valued continuous functions on $X$ is a clean ring. A clean ring is a ...
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25 views

How does the Whitehead quadratic functor act on morphisms?

I am trying to understand the universal quadratic functor $\Gamma$ that appears on the Whitehead exact sequence (J. Whitehead, A certain exact sequence, Annals of Mathematics 52 (1950)) particularly ...
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28 views

What is this product called?

Let $X$ be a finite set and let $2^X$ be its power set. Let $Z$ be some ring (e.g. the complex numbers; it doesn't matter). Suppose $f:2^X\to Z$ and $g:2^X\to Z$ are two functions from $2^X$ to $Z$. ...
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1answer
43 views

Same kernels for homomorphisms of free modules

Let $f: R^n \rightarrow R^m$ be an isomorphism of free $R$-modules ($R$ commutative with unity) and $\pi_1: R^n \rightarrow R^n/\mathfrak m^n$, $\pi_2: R^m \rightarrow R^m/\mathfrak m^m$ the canonical ...
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Problem regarding Cauchy's Theorem.

Let $G$ be a group and define $R_n(G) \subseteq G^n,R_n(G)=\{(g_1,g_2,...,g_n):g_1g_2...g_n=e\}$. $1$. Show that the map $C:G^{n-1} \to R_n(G), ...