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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If $a+bi$ is in $E_k$ then $a-bi$ is also in $E_k$?

I'm currently studying the properties of the Motzkin sets $E_k$, $k\in\mathbb{N}\cup\{0\}$ of the ring $\mathbb{Z}[i]$. The definition of $E_k$ is as follows: $E_0=\{0\}$, $E_1=$units of ...
2
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1answer
33 views

Do we have $\mathbb{C}[T] = \mathbb{C}[\Lambda] = \oplus_{\lambda \in \Lambda} \mathbb{C}[\lambda]$?

Let $\mathbb{C}[T]$ be the coordinate ring of a torus $T$. Suppose that $T$ acts on some variety $X$. Then $T$ acts on $\mathbb{C}[X]$: $t(f) = \lambda(t)f$ ($f$ is a homogenous function on ...
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1answer
24 views

Number of subgroups and normal subgroups

I am struggling to understand how to calculate the nunmber of subgroups with permutations, for example: How many normal subgroups does S3 have? How many subgroups of order 4 has group S4? And does ...
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0answers
18 views

Conditions equivalent to Noetherianness

Let $R$ be a left Noetherian ring. We know that any direct sum of injective left $R$-modules is again injective. Since any injective module is quasi-injective, we infer that (1):"any direct sum ...
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0answers
27 views

Algebra and homomorphism

Is there a homomorphism between each pair of algebras of the same type? Is there an infinite algebra that has only one subalgebra?
3
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1answer
45 views

Find orbit of $1$ for $\sigma$

$\sigma = \left( \begin{array}{cc}1&2&3&4&5&6\\3&1&4&5&6&2\end{array}\right)$ $ 1 \mathop{\rightarrow}^{\sigma} 3 \mathop{\rightarrow}^{\sigma} 4 ...
1
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1answer
40 views

If $ f:G \to H $ is a homomorphism between groups $ G$ and $ H $ and if $ X \subseteq G $ then $ f (\langle X \rangle) = \langle f (X) \rangle $.

Let $\langle X \rangle$ denote the subgroup generated by $ X. $ If $f: G \to H$ is a homomorphism between groups $G$ and $H$ and if $ X \subseteq G $ then $ f (\langle X \rangle) = \langle f (X) ...
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1answer
32 views

Permutation to a power $\sigma^{100}$

$\sigma = \left( \begin{array}{cc}1&2&3&4&5&6\\3&1&4&5&6&2\end{array}\right)$ I need to calculate $\sigma^{100}$ $\sigma = (1,2,3,4,5,6)$ has order 6, and ...
1
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1answer
20 views

Semisimple module example

I need to find an example of a module over $\mathbb{F}[x]$ which is two dimensional over the field $\mathbb{F}$ and not semisimple. I do not know how to do it. Thanks
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1answer
22 views

Show the union of two subrings is generally not a subring

Show that the union of two subrings is a subring if and only if either of the subring is contained in the other. I have no trouble in going from right to left but cannot seem to be able to go from ...
0
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1answer
21 views

compute $\left|<\tau^2>\right|$ for the given permutation

$\tau = \left( \begin{array}{cc}1&2&3&4&5&6\\2&4&1&3&6&5\end{array}\right)$ I need to compute $|\langle \tau^2\rangle|$ I know $\tau^2 = \left( ...
3
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1answer
51 views

Polynomial with n real roots

Let $P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_{1}x + 1$ where $a_i$ are nonnegative and real. Assume $P$ has $n$ real roots. Prove $P(2) \geq 3^n$. I thought I had a good idea about ...
7
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3answers
114 views

When are two direct products of groups isomorphic?

I was thinking about the following problem: Suppose that $G_1 \cong G_2$ are isomorphic groups. Under what conditions on the groups $H_1,H_2$ will we have $$G_1 \times H_1 \cong G_2 \times H_2 ?$$ ...
1
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1answer
33 views

Simple and semi-simple over $\mathbb{Z}$

What is a necessary and sufficient condition on an integer $n$ for $\mathbb{Z}/n \mathbb{Z}$ to be simple as a module over $\mathbb{Z}$? Semisimple? In the case of simple I think that because it is a ...
1
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1answer
32 views

Showing that $x \ast y := x + y - \lfloor x + y \rfloor$ defines a group structure on $[0, 1)$

Define $x\ast y = x+y - \lfloor x+y \rfloor$ where $\lfloor \cdot \rfloor$ is the floor function. How do I prove that $([0,1),\ast)$ is a group? I was trying to separate cases and prove this, but it ...
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2answers
20 views

Find closed form for a sequence using the Fibonacci and Lucas number sequences. [on hold]

Define $A_n$ as follows: $$\begin{align*} A_0 &= 6 \\ A_1 &= 5 \\ A_n &= A_{n - 1} + A_{n - 2} \; \textrm{for} \; n \geq 2. \end{align*}$$ There is a unique ordered pair $(c,d)$ such that ...
5
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1answer
101 views
+50

Galoisgroup of a polynomial

Task: Determine the galois group of $x^4-4x^2-11$ over $\mathbb Q$ The roots are obviously $\pm\sqrt{2\pm\sqrt{15}}$ But I have problems in checking whether just $\sqrt{2+\sqrt{15}}$ is generating ...
1
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3answers
47 views

Let $G$ be a group and $u \in G$ be a fixed element. By the following, prove that $(G,\bullet)$ is a group.

Let $G$ be a group and $u \in G$ be a fixed element. Define the operation $\bullet$ on G as $\forall a,b \in G, a \bullet b=au^{-1}b.$ Prove that $(G,\bullet)$ is a group. So, I know that in order ...
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1answer
23 views

Primitive polynomials: some statements to (dis-)prove

Prove or disprove: i) The sum of primitive polynomials in $\mathbb{Z}[x]$ is primitive ii) The product of primitive polynomials in $\mathbb{Z}[x]$ is primitive iii) There is only a ...
2
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1answer
24 views

Generated field is the same as composite field

I am considering Exercise 13.2.6 from Dummit & Foote's Abstract Algebra: Prove directly from the definitions that the field $F(\alpha_1, \alpha_2, \dots, \alpha_n)$ is the composite of the ...
4
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2answers
120 views

$\mathbb C$ isomorphism to $\mathbb{R} \times \mathbb{R}$ under multiplication

How can I show, that $(\mathbb C,\cdot)$ is not isomorphic to $(\mathbb{R},\cdot) \times (\mathbb{R},\cdot)$ under multiplication? I tried to point out that $f(1) = 1$, then pair $(1,1) \rightarrow ...
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1answer
69 views

The diophantine equation $y^2=x^3+7$ has no solutions.

In my lecture notes there is the following example: The diophantine equation $y^2=x^3+7$ has no solutions. Proof: If the equation would have a solution, let $(x_0, y_0)$, $y_0^2=x_0^3+7$, then ...
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0answers
46 views

$a^2+ab+b^2=c^2+cd+d^2$ prove that $a+b+c+d$ is a composite number for positive integers $a,b,c,d$ [on hold]

(Positive integers $a,b,c,d$ meet this condition $a^2+ab+b^2=c^2+cd+d^2$ )prove that $a+b+c+d$ As in the topic my proof looks like that; $(a+b)^2 - ab=(c+d)^2-cd$ $(a+b)^2 - (c+d)^2=ab-cd$ ...
2
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0answers
22 views

Root system of a simple lie algebra is irreducible

The proposition is from Humphreys. I don't understand how to prove the highlighted statements. How can I express a general element of K? I tried using Cartan decomposition of L but it doesn't work. ...
0
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1answer
12 views

Associative Binary Operation from associative Binary Operation

if $\Delta$ is an associative composition(Binary Operation) on $\mathbb{E}$ and if $a\in \mathbb{E}$, then the composition $\Omega$ on $\mathbb{E}$ defined by $x\Omega y=x\Delta a\Delta y$ is ...
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2answers
26 views

Is there a way to show that the following nonzero polynomials do not divide each other?

Is there a way to prove that the following polynomials $f,g,h$ do not divide each other in the polynomial ring $\mathbb{C}[x,y,z]$? $$f(x,y,z)=x^5-yz, \ \ g(x,y,z)=y^2-xz, \ \ h(x,y,z)=z-xy$$
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2answers
50 views
+100

Number of homomorphisms from one group to another of the same order: G to Z_4

I have $(G,*)$ where $G=\{a_0,a_1,a_2,a_3\}$ and $*:a_i*a_j = a_{(i+j)(mod4)}$ I already showed this is isomorphic to $\mathbb{Z_4}$ Now I need to find all homomorphisms from $G \rightarrow ...
2
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1answer
30 views

Algebraic values of the sine function

First question: For which angles $x$ is $\sin(x)$ a real number that can be expressed using only integers, addition, subtraction, multiplication, division and the extraction of $n$th roots? (With ...
3
votes
1answer
79 views
+100

Clarification about the definition of free module

I am reading this notes. Definition 1: Let $R$ be a commutative ring with $1$. Let $S$ be a set. A free $R$-module $M$ on generators $S$ is an $R$-module $M$ and a set map $i:S\rightarrow M$ ...
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1answer
38 views

Mazur's theorem-abelian group of rational points of an elliptic curve

From Mazur's theorem we have the following: If $E |_{\mathbb{Q}}: y^2=x^3+ax+b, a, b \in \mathbb{Z}$ an elliptic curve, then $$E(\mathbb{Q})_{\text{torsion}} \cong \mathbb{Z}/n\mathbb{Z}, \text{ for ...
2
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1answer
40 views

$\sqrt{I}+\sqrt{J}=R$ implies $I+J=R$

Let $R$ be a commutative ring with unity and $I,J$ ideals of $R$. Suppose that $$ \sqrt{I}+\sqrt{J}=R $$ I want to show that this implies $I+J=R$. Take $r\in R$, then I can write $$ r=a+b, $$ for ...
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2answers
54 views

Extension field of $x^3 - 2$ in $\mathbb Q$

I know this has been asked previous on Stackexchange, but I just want to have something clarified: I am supposed to find an extension K of Q having all roots of $x^3 - 2$ such that $[K:\mathbb Q] = ...
4
votes
3answers
58 views

Computing the galois group of $x^3+3x+1$

I want to calculate the galois group of the polynomial $x^3+3x+1$ over $\mathbb Q$. And I am struggling in finding the roots of the polynomial. I only need a tip to start with. Not the full solution ...
2
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1answer
33 views

find about center of G s.t H is normal subgroup of order 2

Let G be a finite group and H be normal subgroup of order 2. Then order of center of G is 0 1 Even integer $\ge $2 Odd integer $\ge $3 I tried this problem by taking G as $S_3$ and H as $ A_3$, ...
2
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2answers
56 views

Kernel and Image of a group homomorphism

let $G$ be a multiplicative group of non-zero complex analysis.consider the group homomorphism $\phi:G\rightarrow G$ defined by $\phi(z)=z^4$. 1.Identify kernel of $\phi=H$. 2.Identify $G/H$ My ...
1
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1answer
29 views

How can I prove this about the tangent line formula??

The equation of a tangent line to $f(x)$ at $x = t$ is $y = f'(t)(x - t) + f(t)$. Recently, I heard that it is also determined by the remainder of polynomial division of $f(x)$ by $(x-t)^2$. For ...
0
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0answers
39 views

Combinatorial approach to calculate determinant

Suppose you have set of $n*n$ matrices with entries from the set $\{1,-1\}$. Then what can be the maximum determinant which you can obtain from such type of matrices.
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1answer
45 views

Ring with many one-sided zero-divisors

Does there exist a ring all of whose elements are left zero-divisors but only one element is a right zero-divisor? The motivation for asking this question is that if there exists atleast one left ...
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0answers
48 views

Why prime avoidance lemma allows only at most 2 non-prime ideals?

Why prime avoidance lemma allows only at most 2 non-prime ideals? The following is the last part of the proof taken from wikipedia: For the case $n > 2$, choose $z_i \in E \cap (I_i - \cup_{j ...
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5answers
561 views

Studying math all day and really young [closed]

I am very young and want to learn algebra and calculus for fun. What should I keep in mind when I start learning? I am going to try the textbooks I have borrowed out: Dummit and Foote and Spivak's ...
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2answers
33 views

Congruence definitions equivalence

We say that $x$ is congruent to $y$ modulo $z$ when $$x\equiv y\pmod z \iff x \pmod z = y \pmod z$$ Another definition is $$\quad x \equiv y \pmod z \iff \exists k \in \mathbb{Z}: x - y = k z$$ Why ...
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1answer
43 views

Two subgroups $H$ and $K$ of permutation group($G$).such that $H$ is normal in $K$ and $K$ is normal in $G$ but $H$ is not normal in $G$

Is there exist any subgroups $H$ and $K$ of permutation group($G$).such that $H$ is normal in $K$ and $K$ is normal in $G$ but $H$ is not normal in $G$?
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0answers
27 views

e Online source for alternative proofs

I'm looking for some alternative proofs for various theorems. My goal is to compile a list of various proofs each relating to a specific theorem (such as the triangle inequality, Fermat's Little ...
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2answers
49 views

Cremona group of $\mathbb{P}^n$

I know that the complex conjugation $\tau: \mathbb{P}^n \mapsto \mathbb{P}^n$ that sends any point $x$ to the point with complex conjugate coordinates $\tau(x)$ is a homeomorphism. In order to show ...
4
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0answers
22 views

Nilpotent Jacobson radical of $End(M)$

If $M$ is a Noetherian injective left $R$-module, is it true that the Jacobson radical $J=J(End(M))$ of the endomorphism ring of $M$ is nilpotent? I know that if a left $R$-module is Artinian or ...
5
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1answer
40 views

A free direct sum of a projective module

I want to prove that a left module $_RP$ is a projective generator if and only if a direct sum of (copies of) $P$ is free. My try is, first, to observe that $P$ is a generator if and only if for ...
2
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1answer
39 views

Let R be a ring with unity $1_R$ and let S (with unity $1_S$) be a subring of R. Prove that either $1_S=1_R$ or $1_S$ is a zero divisor of R.

Let R be a ring with unity $1_R$ and let S (with unity $1_S$) be a subring of R. Prove that either $1_S=1_R$ or $1_S$ is a zero divisor of R. My attempt: Let $a \in S$. Then $1_S*a = a$ and this a ...
0
votes
2answers
35 views

Change of Basis for $2\times2$ matrix

Suppose I have the matrix basis $\begin{bmatrix}1&0\\0&0\\\end{bmatrix}$ , $\begin{bmatrix}0&1\\0&0\\\end{bmatrix}$ , $\begin{bmatrix}0&0\\1&0\\\end{bmatrix}$, ...
0
votes
1answer
30 views

Vector spaces and multiplicative inverse?

Do vector spaces have multiplicative inverses? They seem to be monoids under $+,\times$, so monoids $(\Bbb F, +)$ and $(\Bbb F, \times)$ where $\Bbb F=\Bbb R \,or\, \Bbb C$ And it is even a group ...
4
votes
2answers
53 views

Example of a Non-Commutative Division Ring With Finite Characteristics

Wedderburn's Little Theorem says that every finite Division Ring is commutative. What is about an infinite Division Ring with prime characteristics? Is this also a Field?