Tagged Questions

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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1answer
4 views

Can you find the number of people at this party?

At a party everyone was shaking hands with others. In all, there were 66 handshakes. Now find the number of people at this party.
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0answers
19 views

homomorphism inducing monomorphism on some quotient group

Let $f:G\rightarrow H$ be a group homomorphism such that $f_* :G_{ab}\rightarrow H_{ab}$ is an isomorphism.. Question is to prove that this induced a monomorphism ...
0
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1answer
21 views

algebra of polynomials of non-commuting variables

In Hatcher's book algebraic topology page 496-497: the Steenrod algebra $\mathcal{A}_2$ is the algebra over $\mathbb{Z}_2$ formed by polynomials of non-commuting variables $Sq^1,Sq^2,\cdots$ modulo ...
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0answers
15 views

Find a complex number for the splitting field

Let $E$ the splitting field of $x^3-2 \in \mathbb{Q}[x]$. Applying the algorithm of the proof of the Primitive element theorem, find a complex $c$ with $E=\mathbb{Q}(c)$. $$$$ I have done the ...
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0answers
9 views

find $f(x) \in \Bbb Z[x]$ s.t $f(x)-\frac {p(x)}{x^k} \in P=\{\frac {(x^2+1)q(x)}{x^k}| k\geq 0, q(x) \in \Bbb Z[x]\}$ where $p(x)\in \Bbb Z[x]$.

Given $\displaystyle\frac {p(x)}{x^k}$, find $f(x) \in \Bbb Z[x]$ s.t. $f(x)-\displaystyle\frac {p(x)}{x^k} \in P=\{\frac {(x^2+1)q(x)}{x^k}\mid k\geq 0, q(x) \in \Bbb Z[x]\}$ where $p(x)\in \Bbb ...
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0answers
15 views

Center of a ring projective?

If $R$ is a ring and $Z(R)$ denotes $R$'s centre, then when is $R$ projective as an $Z(R)$-module?
3
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1answer
23 views

Universal property of the algebraic closure of a field

At page 4 of Strom's "Modern Classical Homotopy Theory" there is a universal formulation of the algebraic closure of a field. You can read it here from google books. Exercise 1.2a is then to convince ...
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2answers
25 views

Set notation query

What do square brackets mean next to sets? Like $\mathbb{Z}[\sqrt{-5}]$, for instance. I'm starting to assume it depends on context because google is of no use.
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1answer
20 views

Global dimension of the center

Let $R$ be a ring. Must the global dimension of the centre $Z(R)$ of the ring $R$ always be atmost that of $R$ itself? I mean is it generally true that: $D(Z(R)) \leq D(R)$ (where D is the global ...
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1answer
93 views

can anyone give a proof by definition :11 is prime in $ \mathbb{Z}[\sqrt{-5}] $

what i did is: assume $\alpha \notin (11),\beta\notin (11), \alpha\beta \in (11)\Rightarrow\exists \gamma, s.t.$ $ \alpha\beta = 11 \gamma$, $\Rightarrow N(\alpha)N(\beta) = 11^2N(\gamma) $ then ...
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0answers
47 views

Irreducibility over Q doesn't imply irreducibility over R

I want a counterexample of polynomial that is irreducible over $\mathbb Q$ but not irreducible over $\mathbb R$ (i.e not maximal over $\mathbb R$).
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1answer
103 views

$|G| + \frac{|G|}{\left|\langle a\rangle\right|} + \frac{|G|}{\left|\langle b\rangle\right|} + \frac{|G|}{\left|\langle ab\rangle\right|}$

Show that for every finite group $G$ and for every elements $a, b \in G$ the following expression $$ |G| + \frac{|G|}{\left|\langle a\rangle\right|} + \frac{|G|}{\left|\langle b\rangle\right|} + ...
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2answers
37 views

Show that $G' = \bigcap_{C\subseteq N \triangleleft G} N$

I am asked to show that $G' = \bigcap_{C\subseteq N \triangleleft G} N$, where $G'$ is the commutator subgroup of $G$, and $C :=\{aba^{-1}b^{-1}\mid a,b\in G\}$. Showing $\bigcap_{C\subseteq N ...
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1answer
15 views

degree of extension of rationals adjoin infinitely many roots

Consider a radical extension of $\mathbb{Q}$ in which infinitely many numbers are adjoined. Each of these numbers is the $n^{th}$ root of some integer, where $n$ can vary. Can any information be ...
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1answer
23 views

What are some examples of these kinds of commutative semirings?

What are some examples of commutative semirings such that the following hold? Multiplication is idempotent i.e. we have $xx=x$ for all elements $x$. Addition is not idempotent i.e. there is at least ...
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0answers
22 views

What do we mean by a group geometrically? [on hold]

What do we mean by a group geometrically? Can we study algebra geometrically? If so, give some articles or books regarding this..
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0answers
16 views

Cyclotomic polynomial identity [duplicate]

STATEMENT Show that $$\Phi_n(X)=\prod_{d\mid n}(X^{n/d}-1)^{\mu(d)}$$ where $\Phi_n(X)$ is the n-th cyclotomic polynomial. QUESTION Could anyone offer any help on how to show this result.
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0answers
8 views

algebraic extension by polynomials roots

Using solubility by radicals, Is possible to prove that $\mathbb{Q}(R_{\xi}) \neq \overline{\mathbb{Q}}$ where $R_{\xi}$ is a set of the unit root. I'm trying to generalize this, i.e., show that if ...
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2answers
30 views

$\mathbb{Q} \simeq \mathbb{Q}^*_+$ isomorphism [duplicate]

Is it true that $(\mathbb{Q},+) \simeq (\mathbb{Q}^*_+, \times)$? If yes then is there any constructive isomorphism?
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2answers
25 views

If $M$ is a $R$-module with $R=R_1\times \cdots \times R_n$ then $M\simeq M_1\times \cdots \times M_n$ where $M_i$ is a $R_i$-module?

Let $R_1, \ldots, R_n$ be rings and consider the ring $R:=R_1\times \cdots\times R_n$. How can I show every $R$-module $M$ is isomorphic to a product $M_1\times \cdots\times M_n$ where each $M_i$ is a ...
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0answers
19 views

How does one actually take the dual of a chain complex?

I know the following about the chain complex used for computing the homology groups of the torus $S^1 \times S^1$: The complex is $0 \to^{\delta_3} \mathbb{Z}[U] \oplus \mathbb{Z}[L] \to^{\delta_2} ...
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1answer
37 views

Prove that any subfield of $\Bbb R$ contains $\Bbb Q$

Prove that any subfield of $\Bbb R$ must contain $\Bbb Q$. Now for any subfield $F$ of $\Bbb R$, $1\in F$ so, $\Bbb Z \subset F \Rightarrow \Bbb Q \subseteq F$. Have I done it correctly?
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1answer
34 views

Determining Lie algebras from commutative diagrams of exact sequences

Suppose that we have the following commutative diagram of graded Lie algebras $$\begin{array} A 0& {\longrightarrow} & C_n & {\longrightarrow} & A_{n+1} &{\longrightarrow} & ...
3
votes
1answer
63 views

Example of $aH \subsetneq Ha$

Problem. Is there an example of a group $G$, a subgroup $H$ and an element $a \in G$ such that $|G : H| < \infty$ and $aH \subsetneq Ha$?
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1answer
40 views

Explanation of Proof that field sum of more than 2 elements is 0. [duplicate]

"Suppose the field $F$ is finite. If $f\colon F\to F$ is any bijection, then we can conclude that $\sum_{x\in F}x=\sum_{x\in F}f(x)$. Let $\alpha\in F$ such that $\alpha\ne 0$. Then $x\mapsto \alpha ...
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0answers
37 views

Finding the group structure of a finite ring

Trying to construct an example I built up this finite ring: $$B=\mathbb{Z}/9\mathbb{Z}[x,y,z,w_1,w_2]/(x^3-1,y^3-1,(x-1)(z+3w_1),(y-1)(z+3w_2),w_1^2,w_2^2,z^2)$$ I need to know the structure of ...
1
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0answers
19 views

Galois group of $E/K$ & Galois group of the extension $E$ over Fixed field?

I've proved this result for myself, but I have doubt in my proof whether it is true : Let $E/K$ be a field extension and $G(E/K)$ its Galois group. Suppose $E^{G(E/K)}$ is its Fixed field, i.e. ...
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1answer
37 views

Exact sequence induces exact sequence

Consider exact sequence $N\xrightarrow{f} G\xrightarrow{g} Q\rightarrow 0$ Question is to prove that this gives exact sequence $N/[G,N]\xrightarrow{\bar{f}} G/[G,G]\xrightarrow{\bar{g}} ...
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2answers
41 views

Let M and N be normal subgroups of a group G such that $G = MN.$ Prove that $G/(M \cap N) \simeq G/M \times G/N$.

Let M and N be normal subgroups of a group G such that $G = MN.$ Prove that $G/(M \cap N) \simeq G/M \times G/N$. Claim 1: $M\cap N$ is a normal subgroup of G: Proof: $1_G \in M$ and $N$ since M ...
5
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2answers
81 views

Product of all elements of a finite group with an unique element of order 2

Well be with you, gentlemen. I have the following problem from Aluffi's Algebra: given a finite group $G$ with an unique element $f$ of order $2$, show that \begin{equation} \prod_{g\in G}g=f ...
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2answers
30 views

Infinite Suzuki Groups

Often I found myself on a symbol like $Sz(F)$ where $F$ is an infinite field. What is the definition of an infinite Suzuki group? Are they linear groups? Where I could find some informations about ...
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0answers
49 views

what does this image describes about mathematics [on hold]

here I am actually finding the abstract mathematical structures and I got this one also which I did not understand.
3
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1answer
42 views

Find a non-abelian group $G$ and a positive integer m such that for all $g,h \in G$, $(gh)^m = g^mh^m$and $(gh)^{m+1} = g^{m+1}h^{m+1}.$

Find a non-abelian group $G$ and a positive integer m such that for all $g,h \in G$, $$(gh)^m = g^mh^m$$ and $$(gh)^{m+1} = g^{m+1}h^{m+1}.$$ I can't find an example.
3
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1answer
40 views

Show that a group is abelian.

Let G be a group and m be a positive integer. Suppose that for all $\alpha, \beta \in G$, $$(\alpha \beta)^m = \alpha^m \beta^m,$$ $$(\alpha \beta)^{m+1} = \alpha^{m+1}\beta^{m+1},$$ and $$(\alpha ...
3
votes
1answer
55 views

Is the following group either a quaternion group or $D_8$?

Let $|G|=2^n$ and $Z(G)=G'=\Phi(G)$ where $\Phi(G)$ is the Frattini subgroup and $|Z(G)|=2$. Is $G$ necassarily either a quaternion group or $D_8$?
2
votes
1answer
21 views

Are (Z/21Z)* and (Z/13Z)* isomorphic? Find an isomorphism between these two group if they are.

Are $(Z/21Z)^*$ and $(Z/13Z)^*$ isomorphic? Find an isomorphism between these two group if they are. $(Z/21Z)^*$ and $(Z/13Z)^*$ are the multiplicative integer modulo group. I have tried ...
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vote
1answer
10 views

The Weyl group and eigenspaces

Let $V$ be a representation of the Weyl group. For any reflection $\sigma_{\alpha}$ (where $\alpha$ is a root), we know that $V$ has two eigenspaces with eigenvalues $1$ and $-1$. The ...
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votes
1answer
23 views

Represent a bijection using a permutation

Let $X = \{1, 2, 3, 4, 5, 6, 7\}.$ For every $n \in X$, write $n^2 - 3n^5 = 7q_n + r_n, 1 \leq r_n \leq 7.$ Define a function $f: X \to X$ by $f(n) = r_n.$ (a) Find an element $\alpha \in S_7$ that ...
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0answers
14 views

The fix points of the Möbius transformations are the eigenspace of a certain matrix.

Let $M=\left( \begin{array}{ccc} a & b \\ c & d \\ \end{array} \right) \in GL_{2}(\mathbb{C})$ and we recall that the Möbius transformation attached to M is the map: $z \to ...
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0answers
12 views

Show that K is a normal subgroup of H H= [a,b;0,d] K= [a,b;0,1] [on hold]

Show that if H= [a,b;0,d] K= [a,b;0,1] then K is a normal subgroup of H.
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vote
1answer
38 views

Lemma 5.2.5 in Springer's Linear Algebraic Groups

I'm stuck trying to understand the first paragraph of this proof. Let $X\rightarrow Y$ be a dominant morphism of affine varieties and denote $B=k[X]$,$A=k[Y]$. Assume there exists $b\in B$ such that ...
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1answer
27 views

A group ring has finite length

I have been given a version of Maschke's Theorem to prove: Let $k$ be a field and $G$ a finite group s.t. $|G|$ is non-zero in $k$. Show that the group ring $k[G]$ is a semi-simple ring. A hint ...
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1answer
31 views

Using the First Isomorphism Theorem / Fundamental Homomorphism Theorem

Here's an exercise on my homework assignment: Let $G = (\mathbb{Z} \times \mathbb{Z}) \times \mathbb{Z}$ under componentwise addition. Let $D$ be the cyclic subgroup of $\mathbb{Z} > \times ...
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2answers
19 views

The field closure of a countable union of countable fields is countable?

If $ K_1 \subset K_2\cdots \subset K_n \subset \cdots$ is a tower of countable fields then their union $ \bigcup_n K_n$ is a countable field. If $\{K_a\}$ is a countable family, but not a tower, ...
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3answers
88 views

The algebraic set is irreducible

If $V$ is an algebraic set of $K^n$, I want to show that $V$ is irreducible exactly when $I(V)$ is a prime ideal. That's what I have tried: We suppose that $V$ is not irreducible. Then, it can be ...
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1answer
22 views

polynomial algebras and their coefficients in prime fields

According to the definition of the polynomial algebras $A(n)$ and $A(n,m)$ for $ n \in \mathbb {N} $ and $ m \in \mathbb {N}^n$, if $\mathbb{F}$ be field $GF(2)$ and $ X_1,...,X_n$ be $n$ pairwise ...
0
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1answer
21 views

Find the order of the point

I want to calculate the order of the point $M$ at the curve $f(x,y) \in \mathbb{R}[x,y]$, when: $$M(0,1), f=(x^2-1)^2-y^2(3-2y)$$ That's what I have tried: $$f(x,y)=x^4-2x^2+1-3y^2+2y^3$$ ...
2
votes
1answer
37 views

Socle of submodule relative to the module

in these notes i am reading i am told that the socle of $K$ (where $K \subset M$ , and $M$ is a module) is = $K \cap$ Soc $ M$ But why is this? i see the intuition but cannot formalize a proof ...
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vote
2answers
50 views

Show that $V(I \cap J)=V(I) \cup V(J)$.

Let $I$, $J$ ideals of $K[x_1, x_2, \dots , x_n]$. I want to show that $$V(I \cap J)=V(I) \cup V(J)$$ I tried the following: $$\subseteq: $$ Let $x \in V(I \cap J)$. From the definition of $V$: ...
5
votes
2answers
46 views

Prove in any integral domain, if $a^2=b^2$ then $a=\pm b$

Prove in any integral domain, if $a^2=b^2$ then $a=\pm b$ An integral domain is a commutative ring with unity having the cancellation property. I don't see how I can use this in proving the ...