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Questions tagged [abstract-algebra]

Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, and modules, among other topics. Whenever you use this tag, please also include related tags like group-theory, ring-theory, modules, etc., in order to clarify which topic of abstract algebra is most related to your question and also to help other users find similar questions through the search engine.

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a lemma in Waerden's algebra,page 213

If a field $P$ is a well odered and if a polynomial of degree $n$ and $n$ symbols $\alpha_1,....,\alpha_n$ are given , tjen a field $P(\alpha_1,....,\alpha_n)$ in which $f(x)$ splits completely into ...
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Unsure about how to prove this

Prove that for all $a,b\in \mathbb R$, $a \cdot b=0$ if and only if $a=0$ or $b=0$. I am unsure how to prove this. Any help would be greatly appreciated!
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2answers
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Product ring isomorphism from example 11.6.3 in Artin's Algebra

I am currently reading chapter 11.6 in Artin's Algebra on Product Rings. There's a proposition that says if $e$ is an idempotent element of a ring $S$ and $e' = 1 -e$ then $S \cong eS \times e'S$. I ...
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Functions restricted to $\operatorname{Spec}(k[x,y]/(x^2,xy))$ with $k$ algebraically closed.

Consider functions $g\in k[x,y]$ restricted to $\operatorname{Spec}(k[x,y]/(x^2,xy))$ with $k$ algebraically closed.(i.e. $Spec(k[x,y]/(x^2,xy))\subset \operatorname{Spec}(k[x,y])$ allows such ...
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2answers
55 views

Proofs With Algebraic Axioms

I wanted to check if my proof of this question is sufficient. This is the question: Prove that for all $a \in \mathbb{R}$, $a \cdot 0 = 0$. And the proof: Lets assume $a=a$ which implies $a-a=0$...
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1answer
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If a group $G$ of order $1200$ with action of $X$ and orbit $6$ then the stabilizer contains a subgroup so $H\triangleleft G$

So I was trying to solve the following question: Let $G$ be a group of order $1200$ with action on $X$. Let $a\in X$. We know that the orbit of $a$ is $6$. Prove that the stabilizer of $a$ has ...
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3answers
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If $H\triangleleft G$ and $H\subset K \leq G$ then $H\triangleleft K$

I was wondering if its true to say If $H\triangleleft G$ and $H\subset K\leq G$ then $H\triangleleft K$ I tried to prove it: We need to prove that for every $k\in K$ we get $k^{-1}Hk=H $. But I'...
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0answers
22 views

Blow-up of affine space along subvariety

I was reading about blow-ups along sub varieties recently in Shafarevich's book and have a question concerning it. Let us take a curve $C$ in $\mathbb A^n$ and consider $X=Bl_C\mathbb A^n$. Since $C$ ...
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Factorization of morphism

This is relative construction in Mumford Algebraic Geometry 2 Chpt 1, Sec 7. Let $X$ be a scheme and $R$ is a sheaf of $O_X$ algebra. There is $Spec_X(R)$ scheme over $X$ s.t. for all $X-$scheme $Y$(...
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Elements of finite order in Z.

I'm doing an exercise in my abstract algebra book. According to it the solution to "Find all elements of finite order in $\mathbb{Z}$" is $ \{0, 1, -1\}.$ Since $\mathbb{Z}$ includes $0$, I assume ...
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1answer
22 views

What properties of $R$ does the monoid ring $R[M]$ inherits?

It's known that polynomial rings $R[x_1,...,x_n]$ inherit some properties of a base ring $R$. For example, (due to the wikipedia article on polynomial rings) if $R$ is an integral domain, then so is $...
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Is it true that $\lim_{n \to \infty} {(P(\forall i,j\leq n \text{ } [X_i, X_j] = e))}^{\frac{1}{n}} = P(X_1 \in Z(G))$?

Suppose $G$ is a group. $\{X_n\}_{n = 1}^{\infty}$ is a sequence of i.i.d. random elements of $G$ satisfying the condition that $$\forall H \leq G \text{ } P(X_1 \in H) = \begin{cases} \frac{1}{...
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Group of order $5^k\cdot 8$ has normal subgroups of order $5^{k},5^{k}\cdot2,5^{k}\cdot4$

Let $G$ be a group of order $5^k\cdot 8$. I was trying to prove that there are normal subgroups of order $5^{k},5^{k}\cdot2,5^{k}\cdot4$. I saw the following statement: Let $P$ be a $p$-Sylow ...
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Does there exist an intermediate step of completion of the algebraic closure $\bar{\mathbb{R}}$ to get the field $\mathbb{C}$?

We know that completion of the field of rationals $\mathbb{Q}$ is the field $\mathbb{R}$ of reals with respect to usual metric. Next, algebraic closure of $\mathbb{R}$ is the field $\mathbb{C}$ of ...
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1answer
38 views

Understanding why the order of $3+\langle 6\rangle$ in $\mathbb{Z}_{15}/\langle 6\rangle$ is $1$

I'm trying to understand the previous thread (link) which finds the order of $3+\langle 6\rangle$ in $\mathbb{Z}_{15}/\langle 6\rangle$. I was following the logic from that thread. First question is ...
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0answers
25 views

$H$ is a subgroup of $G$, $H\neq G$ then $G\neq\bigcup_{g\in G} H^g$ [duplicate]

If $G$ is finite and $H$ is a subgroup of $G$ s.t $H\neq G$ then $G\neq\bigcup_{g\in G} H^g$, where ($H^g = gHg^{-1}$). I managed to show a bijection $N_G(H)g \rightarrow H^g$, i.e from the right ...
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Finding the generators for $\mathfrak{sl}(3,\mathbb{C})$

Hi I am wondering if anyone can help me to figure out the generators\relations for $\mathfrak{sl}(3,\mathbb{C})$? or how i would do this for $ \mathcal{S}_3$? Any help would be much appreciated. ...
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37 views

Every group of order 399 is abelian and not simple

Let $G$ a group with order $399$ = ${3*7*19}$, from Sylow-theorem, I know $n_{3}$ must be $1$ or $7$ or $19$ or $133$, for $n_{7}$ must be $1$ or $57$ and for $n_{19}$ must be 1. I know $n_{19}$ is $1$...
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1answer
50 views
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1answer
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Inequality $\deg(f-g) \le \max(\deg(f),\deg(g))$

$f, g$ polynomials over $\mathbb{C}$. In my book written that $\deg(f-g) \le \max(\deg(f),\deg(g))$ is FALSE. Why? I can't find any counterexample, maybe somebody can explain?
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2answers
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Integral powers of an Element of a group

Q. 1: If the element $a, b$ and $ab$ of a group are each of order $2$, prove that $$ab = ba.$$
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0answers
8 views

Bound on the degree of a polynomial solution to a parametrized equation

Let $K$ be a field, $F = K(T)$ a rational function field over $K$. Let $G \in F[C, X_1, \ldots, X_n]$ be a polynomial with coefficients in $F$. We can consider the polynomial $G(c, X_1, \ldots, X_n)$ ...
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2answers
20 views

Naturally isomoprhism between $S^k(V^*)$ and polynomial ring explanation

I am reading page 74 which explains: Let $S^k(V^*)$ be the space of symmetric $k$-linear functions on a finite dimensional vector space $V$. Then there is an isomoprhism to $\Bbb R[x_1,\ldots, ...
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23 views

Let $M_i, i \in I$, be R-modules. Show that the (external) direct product $\prod_I M_i$ satisfies the following universal property

Let $M_i, i \in I$, be R-modules. Show that the (external) direct product $\prod_I M_i$ satisfies the following universal property relative to the R-homomorphisms $\pi_j:\prod_I{M_i} \to Mj$ defined ...
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1answer
55 views

Degree of splitting field for $f(x) = x^4 - x^2 + 4$ over $\mathbb{Q}$

I started for finding the roots of the polynomial (4 in total) which took the forms $$ \pm \sqrt{\frac{1}{2}\left( 1 \pm i \sqrt{15} \ \right)} $$ I figured that adjoining the positive square root ...
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1answer
24 views

$(S_f)_0$ is a finitely generated algebra if $S$ is. [duplicate]

Let $A, S$ be commutative rings with identity, and assume $S$ is a finitely generated $\mathbb{Z}^{\geq 0}$-graded $A$-algebra. If $f\in S$ is a homogeneous element of positive degree, $S_f$ is a $\...
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0answers
28 views

Is there a connection between Weyl chambers and Lie algebras?

Hi I am currently studying the connection between Weyl groups and Lie algebras. I have established that connection but wondered if there is a connection with the Weyl chambers at all?
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2answers
60 views

Ideal of $\mathbb{Z}[x]$ generated by $3$ and $x^2+1$ is proper

Let $I$ be the ideal of $\mathbb{Z}[x]$ generated by $3$ and $x^2+1$. Show that 1) $I$ is proper, 2) $\phi_a(I)=\mathbb{Z}$ for all $a\in\mathbb{Z}$, where $\phi_a(I):=\{f(a)\mid f\in I\}$ (...
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1answer
21 views

$\text{Tor}_0^R(M,N)\simeq M\otimes_R N$

I'm trying to understand why $\text{Tor}_0^R(M,N)\simeq M\otimes_R N$. Let $\dots\to F_1\to F_0\to M\to 0$ be a free resolution of $M$. Then $$ F_1\to F_0\to M\to 0$$ is exact. Hence $$ F_1\otimes_R ...
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0answers
26 views

Exponential generator of elements of Lie groups

How can it be shown that any element of a Lie group can be represented as $A=e^{ig_A V^A}.$ I think this results from the exponential map. In the case of $SO(3)$ it can be shown through the Taylor ...
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0answers
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Question about a group algebra being not Artinian

Let $F=GF(2)$ and $G=\langle a_1,a_2,\ldots|a_i^3=1,[a_i,a_j]=1\text{ for }i,j\in\mathbb{N}\rangle$, that is the direct product of infinitely many cyclic groups of order $3$. I want to show that the ...
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0answers
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When $\bigcap_{i=1}^n I_i =\prod_{i=1}^n I_i$ for any (noncommutative) ring?

In some thesis there are given ideals $I_i \subset R$ which are pair comaximal and generated by central elements of ring $R$ and it's written "then $\bigcap_{i=1}^n I_i =\prod_{i=1}^n I_i$ for any $n\...
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1answer
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How many countable non-isomorphic models does acfp have?

I know $ACF_p$ is $k$ categorical for all uncountable $k$ but I cant find the nunber of countable non-isomorphic models of it ... I think since the number of prime numbers is countable it should be at ...
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2answers
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Finding the Weierstrass normal form of an elliptic curve.

I am given an elliptic curve with homogeneous equation $X^3+Y^3+Z^3=0$ over a field K and I am asked to find the Weierstrass normal form. I started by noticing that the point $(1,-1,0)$ is in the ...
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2answers
63 views

Proving that sum of two elements is not invertible in a ring

Let $(A,+,.)$ be a ring such that A is not a field and $x^2=x, \forall $ non-invertible $ x\in A $. Prove that: a) $a+x$ is not invertible for all $a,x\in A$ with $a$ invertible and $x\ne0$,$x$ ...
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0answers
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Deriving a formula for the addition of two points in an elliptic curve and finding its Weierstrass normal form.

I am given the elliptic curve with homogeneous equation $E: X^3+Y^3+Z^3=0$ over a field $K$ with a point $P_{\infty}=[1:-1:0]$. I am told that you can turn $E(K)$ into a group with identity element $...
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1answer
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Image of a basis of a free module is a basis of a vector space

Suppose $M$ is a free module over a commutative ring $A$ with unity. Let $N=\mathscr m M$, where $\mathscr m$ is a maximal ideal of $A$. How do I show that the image of any $A$-basis $\{m_i|i=1,\dots,...
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1answer
21 views

Image of a basis of a free module

Suppose $M$ is a free module with basis $\{m_i|i=1,\dots,s\}$ over a commutative ring $A$ with unity. Suppose $N$ is a submodule. Is it true that the image of a basis of $M$ under the map $M\to M/N$ ...
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3answers
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Ideal generated by two elements is maximal in $\mathbb{Z}[x]$

The question is to show that $I = (x^4 + x^3 + x^2 + x + 1, 2) \subset \mathbb{Z}[x]$ is a maximal ideal. I'm familiar with the results that $R / I$ is a field iff $I$ is maximal, and $R/I$ is a ...
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0answers
23 views

Subgroup generated by all elements with odd order [duplicate]

Let $G$ be a finite group, and let $H$ be the subgroup generated by the set of elements with odd order in $G$, we will denote this set as $S$. Prove that $H$ is normal in $G$ and that $G\setminus H$ ...
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27 views

Group displaying abstract presentation

I'm currently looking how it is possible for Weyl groups to be completely determined by their generators and relators. I think I understand this.... but am wondering if anyone is able to provide me ...
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Abstract Weyl group for a Lie algebra

Can someone please help me with how I find the Weyl group (abstractly) from the Cartan Matrix. I am using the $A_2$ root system as my example and so far have the Coxeter matrix for that but unsure ...
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1answer
18 views

simple group with finite index, deriving a precise estimate

Suppose G is a simple group, H its proper subgroup of finite index. The first part of the question was to prove G is finite, which I did by showing it is isomorphic to a subgroup of $S_n$ where $n$ is ...
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2answers
58 views

The set of one-sided inverses is not a group under multiplication

We proved that if $A$ is a ring and $U$ is the set of elements of $A$ which have both a right and left inverse, then $U$ is a multiplicative group. Now I ask for an example to show that the elements ...
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1answer
36 views

No simple groups of order $90$

So I was trying to study the Sylow theorems. I tried to prove that there are no simple groups of order $90$. I proved that if $n_{5}=1$ or $n_{3}=1$ then it has a normal subgroups (meaning $G$ is ...
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6answers
47 views

Does $\mathbb{F}_9$ contain a 4th root of unity?

I realised that I don't know how to construct $\mathbb{F}_9$. I'm guessing that $\mathbb{F}_9 = \mathbb{F_3(\theta)}$, where $\theta$ is the root of some irreducible polynomial over $\mathbb{F}_3[x]$ ...
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1answer
24 views

$\tilde \phi:\Gamma (W)\rightarrow \Gamma(V)$ may not extends to all $k(W)$

A problem from Fulton's Algebraic Curves:-- Let $\phi:V\rightarrow W$ be a polynomial map between two affine varieties and $\tilde \phi:\Gamma (W)\rightarrow \Gamma(V)$ be the induced map between co-...
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0answers
36 views

How to prove that $\dim_k(k[x_1,\dotsc,x_n]/\sqrt{I})=|V(I)|$

How to prove that $\dim_k(k[x_1,\dotsc,x_n]/\sqrt{I})=|V(I)|$. I know that Hilbert Nullstellensatz will be required but I can't it out how?? With the notation common in algebraic geometry, the ...
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2answers
44 views

Finite set $H\subset$ group $G$ is subgroup $\iff H$ is closed under the binary operation of $G$

I have the proof in my textbook but I am just not able to get it. Here's the proof and parts which I don't understand. Main Proof 1. How does showing ah1 = ah2 prove that |aH| = |H| ? I get that ...
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0answers
41 views

Why is $k[x_1] \to A$ not finite and $\phi(y)=x_1+x_2$ finite?

I was reading https://www.math.columbia.edu/~dejong/courses/deJongNotes.pdf. (Observe you don't have to look at the link as the question is self contained) See Example 1 in the beginning. There I ...