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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Subring of a field extension is a subfield

For the first part, I am trying to show that any subring of $E/F$ is a subfield where $E$ is an extension of field $F$ and all elements of $E$ are algebraic over $F$. My solution is to just show an ...
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give a group that is isomorphic to the figure.

I think if I get help with one of these I should be good on the rest. 23) is concerned with figure a) It has one symmetry and 4 possible points. seems like it would have two elements that map to ...
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Associative Binary Operation(composition) is anticommutative iff idempotent…

if Binary Operation, $\Delta$, defined on $\mathbb{E}$ is associative, then $\Delta$ is anticommutative iff $\Delta$ is idempotent and $x \Delta y \Delta x=x$, ∀$x,y \in \mathbb{E}$.
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Lifting map between finitely generated modules

Suppose $A$ is a commutative ring with unit, and $M$ is a finitely-generated module with the surjection $\pi: A^n \twoheadrightarrow M$. Let $f : M \to M$ be a module homomorphism. I am trying to see ...
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What's the difference between these two definitions of polynomial function?

Definition 1: Given $a_n,...,a_1,a_0 \in \mathbb{R}$, a polynomial function is a function $p:\mathbb{R} \rightarrow\mathbb{R} $ such that $p(x)=a_nx^n+...+a_1x+a_0$ Definition 2: The function ...
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Order of the Rubik's cube group

Associated to the Rubik's cube is a group as described in this Wikipedia article: $G = \langle F, B, U, L, D, R\rangle$. For example, the element $F$ corresponds to rotating the front face clockwise ...
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1answer
20 views

I have a question about Viete's formulas

If I have a polynomial $a_n x^n + a_{n-1}x^{n-1}+ \cdots + a_1 x + a_0$, and the roots of the polynomial is $r_1,r_2,\ldots,r_n$, then I can rewrite the polynomial as, $a_n x^n + a_{n-1}x^{n-1} ...
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Does $G\to X$ have dense image iff $T_eG \to T_{\theta(e)}X$ is surjective?

Let $G$ be a connected algebraic group, $X$ a variety and and $\theta : G\to X$ be a morphism of varieties. (In particular it could be the orbit of some action of $G$.) Consider the corresponding ...
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$x^2+3$ has two zeros over ${\Bbb F}_p$ provided that $x^2+x+1\in{\Bbb F}_p[x]$ has two?

The following is an exercise in abstract algebra: If $p=1\pmod{3}$, then $x^2+x+1\in\Bbb{F}_p[x]$ has two zeros. Prove in this case that $-3$ is a quadratic residue mod $p$. Showing that ...
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Question about the ring of polynomials bounded on a real variety

Suppose that $I$ is a prime real radical ideal in the polynomial ring $\mathbb{R}[x_1,\ldots,x_n]$. "Real radical" means that if a sum of squares $a^2+b^2+\ldots$ is an element of $I$ then so are the ...
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Vector Spaces and Simple Modules

Let $G$ be a finite group and let $R = \textbf{R}[G]$ be the group ring of $G$ with coefficients in the field $\textbf{R}$ of real numbers. Let $V$ be an $R$-module which is finite-dimensional as an ...
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35 views

Infinite rings with lots of zero divisors

Today I was trying to find an infinite ring $R$ whose all nonzero and nonidentity elements were zero divisors and actually found one: $\mathcal R =\text{Fun}(\mathbb N, \mathbb Z/2\mathbb Z)$. Given a ...
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Algebra finite over a subalgebra

I'm afraid my question is quite stupid, but I can't find the definition I need: given $A$ a graded algebra, i would like to know what does it mean that $A$ is "finite" over a sub-algebra. In ...
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38 views

Isomorphism of quotient rings of $\mathbb{Z}[x]$

$\mathbb{Z}[x]/(x^2-a)$ is isomorphic to $\mathbb{Z}[x]/(x^2-b)$. Which $a$ and $b$ must be? Of course when $a$ equal to $b$, but I can't find any others. I tried to divide $(x^2-a)$ to ...
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Intermediate field extension

Consider de field $\mathbb F_2$ of two elements. Let $\theta$ be a zero of the irreducible polynomial $t^4+t+1$ and consider the extension $\mathbb F[\theta]$. May question is: Can we find an ...
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18 views

Show that $G$ contains elements $a,b$ s.t. $a^2=b^3=e$ and $ aba=b^2=b^{-1}$

Given that $|G|=6$ and is not commutative. I've showed that generators of $G$ have periods either $2$ or $3$, since $|H_a|=\text{period of generator a}$ divides $|G|$. Since $G$ is not commutative it ...
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28 views

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 ...
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1answer
27 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
20 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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15 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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26 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?
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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 ...
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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
26 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 ...
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1answer
17 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 ...
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1answer
19 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( ...
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47 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 ...
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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 $$G_1 \times H_1 \cong G_2 \times H_2 ?$$ ...
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1answer
31 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 ...
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1answer
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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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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 ...
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1answer
52 views

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

How can I show, that $(\mathbb C,.)$ is not isomorph to $(\mathbb{R},.) \times (\mathbb{R},.)$ under multiplication? I tried to point out that $f(1) = 1$, then pair $(1,1) \rightarrow 1 + 0i$, but ...
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1answer
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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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$a^2+ab+b^2=c^2+cd+d^2$ 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; $(a+b)^2 - ab=(c+d)^2-cd$ $(a+b)^2 - (c+d)^2=ab-cd$ ...
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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. ...
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1answer
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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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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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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 ...
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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 ...
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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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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 ...
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$\sqrt{I}+\sqrt{J}=R$ 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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51 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] = ...
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3answers
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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 ...
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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$, ...