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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Set $A$ not closed under $\star$ then $A$ not a group under $\star$?

I am currently doing some exercises. I have been through some examples of solutions in other books that questioned me. I know well that $(A,\star)$ is a group if it satisfies the following points, ...
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23 views

Invertible element of $S$

Let $S=\mathbb{Z}[\sqrt{2}]$ = {$a+b\sqrt2|a,b\in \mathbb{Z}$} and $R = \mathbb{Q}[\sqrt2]$ = {$\alpha + \beta\sqrt2 | \alpha, \beta \in \mathbb{Q}$}. Consider $x=3+2\sqrt2$ and $y = 3+4\sqrt2$ ...
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1answer
25 views

In a Category, Is the Set of Morphisms Between Objects Defined to Be All Possible Morphisms?

For instance, if I have a category $\mathfrak{M}$ whose objects are families of morphisms $\{f_i\colon A_i\to B\}_{i\in{I}}$, then if we consider two such objects, say $C=\{f_i\colon A_i\to B\}$ and ...
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0answers
22 views

Can we use the Nullstellensatz?

In $\mathbb{C}[x, y, z]$ we have that $V=\{y-x^2, z-x^3)=\{(t, t^2, t^3) | t \in \mathbb{C}\}$. To show that $$I(V(y-x^2, z-x^3))=\langle y-x^2, z-x^3\rangle $$ can we use the Nullstellensatz??
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Find $f_{1}, f_{2}$ in $\mathbb{Z_{6}}[x]$ such that deg$(f_{1})$ = deg$(f_{2}) = 2$ and deg$(f_{1}+f_{2})=1$

Find $f_{1}, f_{2}$ in $\mathbb{Z_{6}}[x]$ such that deg$(f_{1})$ = deg$(f_{2}) = 2$ and deg$(f_{1}+f_{2})=1$ Find $g_{1}, g_{2}$ in $\mathbb{Z_{6}}[x]$ such that deg$(g_{1})$ = deg$(g_{2}) = 1$ and ...
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1answer
26 views

Highest Common Factor

Which is the smallest integer number $a$ so that the highest common factor of $a$ and $a+5$ is not $1$ and the highest common factor of $a$ and $a+7$ is not $1$ either? I think that it is $35$. Am I ...
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38 views

Polynomial ring addition in $\mathbb{Z_{6}}$

I know this is a very simplest question ever. But, I need help with understanding it. So here it goes... Let, $f(x) = \bar{1}+\bar{2}x+\bar{3}x^2$ and $g(x) = \bar{4}+\bar{5}x$ $\in \mathbb{Z_{6}}.$ ...
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1answer
26 views

If $G/Z(G)$ is of size $qp$ and $p-1$ is not divisible by $q$ then $G/Z(G)$ is cyclic?

I have $G/Z(G)$ with size $pq$, $p, q$ are prime and $p>q$; $(p-1) $ is not divisible by $q$ How do I deduce from the above that $G/Z(G)$ is cyclic?
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45 views

Which algebraic intuition can be used in fields

I wonder what basic laws of arithmetic of reals e.g. $x^n y^m = (xy)^{m+n}$ holds for fields. Every time I take some book on abstract algebra it proves very abstract and unpractical properties. So I ...
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23 views

Group Action and “nice” Approximation!

Studying the action of a group $G$ on a set $X$ is naturally the same as looking at the group homomorphism $\alpha: G \rightarrow Perm(X)$. So, for a given group $G$, classifying all sets $X$, on ...
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1answer
23 views

A question about $rad(M)$

We know that for any $R$-module $M$ the radical $rad(M)$ is the sum of all small submodules of $M$. My question: "if $x\in rad(M)$ is it true in general that $Rx$ is a small submodule of $M$? ...
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47 views

Irreducible polynomials of the form $x^n - q$

Is there any easy way to see the following? Let $n\in\mathbb{N}$. Let $a,b\in\mathbb{R}$ s.t. $a < b$. Then, there exists $q\in\mathbb{Q}$ s.t. $a<q<b$ and $p(x) = x^n - q$ is irreducible ...
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0answers
21 views

Find a non zero element $z\in \mathbb Z_{100}$ such that $yz = 0_{R}$ and $zy = 0_{R}$ where $y =\overline{ 14}$

Find a non zero element $z\in \mathbb Z_{100}$ such that $yz = 0_{R}$ and $zy = 0_{R}$ where $y =\overline{ 14}$ For this I have found such an element to be $\overline{50}$ since ...
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19 views

xor of three relations using Relation Algebra operations

Suppose I have three relations R1, R2, and R3. How can I specify xor of these three relations using relation algebra operations. How this scales up (for example, for four relations)? Thanks I add ...
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24 views

…and a and b are relatively prime positive integers. Find a+b. [on hold]

Let $P = \log_a b$, where $P = \log_2 3 \cdot \log_3 4 \cdot \log_4 5 \cdots \log_{2008} 2009$ and $a$ and $b$ are relatively prime positive integers. Find $a+b$.
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1answer
24 views

Why is that the radical ideal?

In my lecture notes we have the following: Definition: $f, g \in \mathbb{C}[x, y]$ $f \sim g \Leftrightarrow \exists c \in \mathbb{C}, c \neq 0$ such that $g=cf$ Example: If $f \sim g ...
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1answer
35 views

Example of a linear algebraic group which is not a Lie group

I am trying to reconcile the notions of algebraic groups, linear algebraic groups, Lie groups, and Lie algebras, along with their notions of root systems, maximal tori, etc. To begin, I am trying to ...
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1answer
77 views

What is an intuitive way to think of Cauchy's theorem?

I am looking at a problem which involves an understanding of why a finite group $G$ has an element with order $p$ if $p$ is a prime factor of $|G|$. I have looked at several resources and proofs ...
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1answer
32 views

Tensor Algebra: Symmetrization & Antisymmetrization

Problem Given the tensor algebra: $$TV:=\sum_{k=0}^\infty{\bigotimes}^k V$$ Regard the symmetrization and antisymmetrization: ...
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1answer
29 views

Find the order of $GL_2(Z_{p^{n}})$ for each prime ${p}$ and positive integer ${n}$.

Let $GL_2(Z_m)$ denote the multiplicative group of invertible $2 * 2$ matrices over the ring of integers modulo m. Find the order of $GL_2(Z_{p^{n}})$ for each prime ${p}$ and positive integer ${n}$.
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Evaluating minimal polynomial over a field $F$ as a characteristic polynomial for a $F$-linear map.

I'm considering $K/F$ to be an extension of degree $n$. I've shown that for any $a\in K$, the map $\mu_a : K → K$ defined by $\mu_a(x) = ax$ for all $x\in K$, is a linear transformation of the ...
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0answers
28 views

Proving that the irreducible polynomial of $\sqrt 2 + \sqrt 3$ over $\mathbb Q$ is irreducible modulo $p$ for every prime $p$. [duplicate]

I've computed the irreducible polynomial of $\sqrt 2 + \sqrt 3$ over $\mathbb Q$ to be $x^4+10x^2+1$. I want to show that this polynomial is irreducible module $p$ for every prime $p$. How do I do ...
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1answer
31 views

An inequality with elementary symmetric polynomials

Fix a natural number $n\geq 1$. Let $a_1, \ldots, a_n$ be $n$ real numbers such that $a_i>0$ for each $i$. Show that for each natural $k$ with $0\leq k\leq n$ $$e_k(a_1,\ldots, ...
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1answer
15 views

$V_1=V(x-y)$ and $V_2=V(x+y)$ are algebraic sets

I am looking at irreducible algebraic sets. $V \subseteq K^n$ is an algebraic set $\Leftrightarrow$ it is of the form $V(I)$, where $I=$radical Ideal of $K[x_1, x_2, \dots , x_n]$. At my lecture ...
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1answer
14 views

Algebraic set - Radical Ideal - $Rad(Rad(I))=Rad(I)$

In my lecture notes we have the following: $V \subseteq K^n$ is an algebraic set $\Leftrightarrow$ it is of the form $V(I)$, where $I=$radical Ideal of $K[x_1, x_2, \dots , x_n]$. It stands that ...
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1answer
11 views

Freely generated modules and bases

I'm trying to show that a subset $S=\{m_1,\dots,m_k\}$ of an $R$-module $M$ generates $M$ freely if and only if $S$ is a basis for $M$. I think I can see the 'if' - using the unique expression of a ...
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1answer
17 views

Determining whether ${p^{n-2} \choose k}$ is divisible by $p^{n-k -2}$ for $1 \le k < n$

Let $p$ be an odd prime and $n \ge 3$ a positive integer. I would like to know whether ${p^{n-2} \choose k}$ is divisible by $p^{n-k -2}$ for $1 \le k < n$. It should be noted that one can ...
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1answer
21 views

Algebraic Set-Radical Ideal-Nullstellensatz

In my lecture notes there is the following: $$I \rightarrow V(I) \rightarrow I(V(I))$$ It stands that in general $I \subsetneq I(V(I))$. The equality stands if and only if $I$ is a radical ...
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1answer
24 views

Isomorphism of finitely generated groups

Let G and H be two groups such that $G=<a,b>$, $H=<c,d>$ and o(a)=o(c), o(b)= o(d). Does that imply that G and H are isomorphic? or some other condition is also required ?
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1answer
21 views

A question about left ideals

Let $V$ be a finite-dimensional vector space over a field $F$. I need to prove that for every left ideal $I$ of $\operatorname{End}_F(V)$, there is only one subspace $W$ of $V$ for which $I$ = {$A ...
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13 views

Definitions of the group of cycles/group of boundaries

first I want to clarify that the class I am referring to is not really about homological algebra, rather about Galoistheory. Still we defined the group of cycles/ the group of boundaries (first ...
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Proof of the proposition $V(S)=V(\langle S \rangle )$

In my lecture notes we have the following: Proposition: $$V(S)=V(\langle S \rangle )$$ Proof: $$\langle S \rangle=\left \{\sum_{i=1}^m g_i f_i | f_i \in S, g_i \in R=K[x_1, x_2, \dots , ...
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1answer
19 views

The generator of a cyclic quotient group

(I'll use (a) to denote the subgroup of G generated by a) Let G be a finite abelian group of order n. Let a be an element of G of order k. We can easily see that (a) is a normal subgroup of G. If we ...
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1answer
30 views

Prove that each of the following sets, with the indicated operation, is an abelian group

$1.$ $x * y = x + y + k$ ($k$ a fixed constant), on the set $\mathbb R$ of the real numbers. $x * y = x + y + k = y + x + k = y * x.$ Commutativity holds. $(x * y) * z = (x + y + k) * z = (x + y + ...
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1answer
22 views

Cyclotomic polynomials and Galois groups

According to this question I want to extend the question from there. Lets consider again the galois extension $\mathbb Q(\zeta)/\mathbb Q$ where $\zeta$ is a primitive root of the $7^{th}$ cyclotomic ...
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1answer
92 views

Do these groups have a meaning?

Let $G$ be a group. We can say that $Aut(G)\leq S_G$ where $S_G$ denotes the set of all bijection from $G$ to $G$. But $S_G$ is not a good bound for $Aut(G)$ as $S_G$ grows very fast. Let's define ...
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26 views

Superfluous radical

If $M$ is a projective Artinian (left) module over a ring $R$, could one say that the radical of $M$ is a superfluous submodule in M? I know that for a projective module $M$ the radical of $M$ equals ...
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29 views

Prove $G\cong H\oplus \Bbb{Z}^{k}$.

Let $G$ be an abelian group and let $H$ be a subgroup. Let $G/H\cong \Bbb{Z}^{k}$. Prove $G\cong H\oplus \Bbb{Z}^{k}$. What I did so far is: there is an epimorphism from $G$ to $\Bbb{Z}^{k}$ such ...
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Name of this theorem? (Generalization of the class equation)

Let $X$ be a nonempty finitr set and $G$ be a finite group acting on $X$. Let $G.x_1,...,G.x_n$ be the distinct orbits of $G$. Define $F(X)=\{x\in X : \forall g\in G, g.x=x\}$. Then ...
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36 views

Ideal as kernel of a homomorphism in Gaußian integers

Consider the ring $\mathbb{Z}[i]$ of Gaußian integers. The principal ideal $(1+i)$ is maximal ideal in this ring. Since ideals are kernels of some homomorphisms, I would like to see a homomorphism ...
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1answer
23 views

Does first isomorphism theorem work both sides?

The theorem says that if I have a group homomorphism, then the kernel is normal and the image is isomorphic to the domain group modulo the kernel. Now, suppose I have $G/K \cong{H}$ where $G$ and ...
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35 views

Proving complete reducibility of modular representations

Let $G$ = $S_{3}$ and consider the $3 \times 3 $ permutation representations. For example, we have $$ \psi (123) = \begin{pmatrix} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\\ ...
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22 views

Operations with ideals: sum and product

Operations at ideals. The sum is defined as $$I_1 + I_2 + \dots + I_m =\{a_1+a_2+\cdots +a_m\mid a_i \in I_i\}.$$ It can be proven that $$I_1 + I_2 + \dots + I_m \trianglelefteq R$$ and each ...
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29 views

$Rad(I)$ is an ideal of $I$

$$Rad(I)=\{a \in R | \exists n \in \mathbb{N} \text{ such that } a^n \in I\}$$ R is a commutative ring, I is an ideal. To show that $Rad(I)$ is an ideal of $I$, we have to show that for $a,b \in ...
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1answer
78 views

Galois theory in reverse

Let $K/F$ be a Galois extension. The fundamental theorem of Galois theory says that there is a bijection between subfields of $K$ containing $F$ and subgroups $H$ of $G$, where $G$ is the Galois group ...
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1answer
23 views

Quadratic form - non-degenerate

(The order of a quadratic form is defined to be the order of the matrix $A$) Definition: $Q(x_1, x_2, \dots , x_n)$ is called non-degenerate $\Leftrightarrow (a) $A=$invertible (b) At each $v \in ...
3
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2answers
44 views

Centre of a group and normalizers

Let $G$ be a group and let $A \subset G$ be a non empty subset of $G$.Define the following subsets of $G$ $$Z(G) = \{z \in G \space | \space zx =xz \space \space \forall x \in G \}$$ $$N_G(A) = \{h ...
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0answers
15 views

Closed slice of affine space has closed projection

Let K be an algebraically closed field and suppose a set $\{ \overline{x} \} \times A$ is closed in the Zariski topology of the affine space $K^m \times K^n$ ($\overline{x} \in K^m, A\subset K^n$). ...
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1answer
10 views

Number of Binary Operations(Compositions) with a specific neutral element

Let $E$ be a finite set of $n$ elements. If $a\in \mathbb{E}$, for how many compositions on $\mathbb{E}$ is is $a$ the neutral element.
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1answer
41 views

Is the way I simplify my notation?

Do you agree with how I simplify this equation notation $$\sum_{i=1}^n \phi_{v_i} \left(\sum_{i=1}^m \phi_{l_i}\right)=\sum_{(i,j)\in \mathcal{S}}\phi_{v_i}\phi_{l_j},$$ where I define ...