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 $\sigma \in Aut(G)$where $|G|$ odd has order 2 there exists Sylow $p$-subgroup with $\sigma(P)=P$

Let $G$ be a group of odd order and $\sigma$ an automorphism of G of order 2. Show that if the prime $p$ divides $|G|$ then there exist a Sylow $p$-subgroup $P$ such that $\sigma(P)=P$.
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38 views

Is algebra over a set also algebra over a field?

During my studies I have come across two different notions of the term "algebra", namely algebra over a set and algebra over a field (the field its vector space always being Euclidean space in my ...
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1answer
48 views

Find all the ideals of $\mathbb Q[X]$

I am trying to find all the ideals of the ring $\mathbb Q[X]$. If $I$ is a non trivial ideal of $\mathbb Q[X]$, then there exists $p(x) \in \mathbb Q[X]$. Since $I$ is an ideal and a group under ...
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0answers
21 views

Finding degree of an extension

Find the degree of the field extension $\mathbb{Q}[\sqrt[3]{2},\sqrt[3]{3}]$ over $\mathbb{Q}$. My approach: Call the desired degree $n$. Clearly, $3|n$ and $n\leq 9$. So possible values of $n$ are ...
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2answers
40 views

$\dim \mathscr L(\mathbb R^m,\mathbb R^n;\mathbb R^p)$

I know we can identify the set $\mathscr L(\mathbb R^m;\mathbb R^n)$ of the linear transformations $f:\mathbb R^m\to \mathbb R^n$ with a matrix $M_{n\times m}$ in the canonical way. Then, we have ...
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1answer
32 views

Left ideals of $M_n(K)$ [duplicate]

Let $K$ be a field and $n \in \mathbb N$. Show the following (i) Let $V \subset K^n$ be a subspace and $I_V$ the subset of $M_n(K)$ consisting of all the matrices whose rows belong to $V$. Prove that ...
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1answer
64 views

Automorphisms of the group of integers $\mathbb Z$

Can anyone help me showing $\operatorname{Aut}(\mathbb Z)\simeq \mathbb Z_2$? I guess I should define an homomorfism $\phi:\mathbb Z\longrightarrow S(\mathbb Z)$ with kernel $2\mathbb Z$ and image ...
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50 views

“Reverse” quotients.

Given a set $A$ and a equivalence relation $\sim$ on $A$, one can construct the quotient $A/{\sim}$. But what about the opposite? More precisely, given a set $X$, is there a way to know whether exists ...
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17 views

Using Conversion Factors to Determine Exchange Rate

While in Europe, if you drive 125km per day, how much money would you spend on gas in one week if gas costs 1.10 euros per liter and your car's gas mileage is 39.0mi/gal ? Assume that 1 euro = 1.26 ...
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65 views

Idempotents in quotient ring

Let $R$ be the ring $\mathbb{Z}[X,Y,Z]/(Y − X + 1, Y − Z + 2, 3X^2 - YZ + 3X +2Y + 4)$. What are the solutions $e \in R$ of the equation $e^2 = e$ with $e \not\in \{0,1\}$? So $e$ has to be a ...
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42 views

Order of $a^m$.

Let $G = \langle a \rangle$ a finite cyclic group of order $n$. Prove that $|a^m| = \frac{n}{\gcd(m,n)} = \frac{\mathrm{lcm} (m,n)}{m}$. I managed "half" of it. Write $|a^m| = k$ and $d = ...
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36 views

What is an easy to read book on category theory including the introduction of some killer apps for the theory? [duplicate]

What is an easy to read book on category theory including the introduction of some killer apps for the theory ?
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1answer
76 views

How many automorphisms does $S_3\times S_3$ have?

I've shown that $|\text{Aut}(S_3\times S_3)|\ge 72$, how can I show that $|\text{Aut}(S_3\times S_3)|\le 72$ ?
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51 views

Any general “formula” solutions for higher order polynomial equation?

We know that fifth (or higher) degree polynomial equation has no general solution formula using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of ...
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1answer
70 views

Infinite primes of a number field

Let $K$ be a number field. I know that to each real and to each complex conjugate pair of embeddings of $K$ there corresponds exactly one prime (equivalence class of absolute values) of $K$. How do I ...
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1answer
38 views

Let $H\le G$ s. t. whenever $Ha≠Hb$ then $aH≠bH$. Prove that $gHg^{−1}\le H\;$ $\forall g\in G$. [closed]

Suppose that H is a subgroup of G such that whenever$ Ha \ne Hb $ then $ aH \ne bH $. Prove that $ gHg^{-1} \subseteq H$ for all g in G.
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37 views

A seemingly simple fact about construction of maps proven categorically, i.e. by universal properties

If I have four sets $A,B,C,D$ and two maps $f_1 : A \to C$ and $f_2 : B \to D$, it is easy to find a unique map $f : A\times B \to C\times D$, namely $$ f(a,b) := (f_1(a), f_2(b)). $$ But now I want ...
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2answers
27 views

Decomposing linear mapping between direct sums of vector spaces

In a textbook I found the following exercise: Let $V = V_1 \oplus V_2 \oplus \cdots \oplus V_n$ and $W = W_1 \oplus W_2 \oplus \ldots \oplus W_n$ and $A_i \in \mbox{hom}(V_i, W_i)$ for $i = 1,\ldots ...
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17 views

Reversible and Directly Finite Rings? [closed]

A ring $R$ is reversible if $ab=0$, then $ba=0$. A ring $R$ is directly finite if $ab=1$, then $ba=1$. Give an example of a non-commutative ring that is reversible? Give an example of a ...
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1answer
33 views

How to show $\alpha=i_L$ and other equalities without using isomorphism?

This is a question of Commutative Algebra.It was given in my class.I was able to solve it to some extend.But I have some doubts. Please help me. Thnx in advance. $A$ is a commutative ring with ...
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2answers
24 views

Difference between External and internal direct product?

What is the difference between External and internal direct product ?? I think both of them boil down to the same thing .
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2answers
38 views

Units in $\mathbb Z[\sqrt{2}]$ [duplicate]

I am trying to find all units in $\mathbb Z[\sqrt{2}]$. Suppose $x=a+b\sqrt{2}$ is a unit. Then there is $y=c+d\sqrt{2}$ such that $$xy=(a+b\sqrt{2})(c+d\sqrt{2})=1$$ So $$ac+2bd+(ad+bc)\sqrt{2}=1$$ ...
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52 views

Zero divisors of $C[0,1]$ [duplicate]

Find the zero divisors of the ring $R=C[0,1]$ the continuous functions $f:[0,1] \to [0,1]$. I could thought of a set $S$ that I think is included in the set of zero divisors, but I am not sure if $S$ ...
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1answer
33 views

Number of Sylow 2-subgroups of a special linear group

Find the number of Sylow $2$-subgroups of the special linear group of order 2 on $\mathbb{Z}$ (modulo $3$). I think it will be $1$. But I failed to prove it using the counting principle. It has $4$ ...
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0answers
20 views

How do I show that both the additive and multiplicative groups of an infinite field are non-cyclic? [duplicate]

I've tried mimicking the proof in case of $\mathbb Q$ to deal with the characteristic zero case, but can't do the characteristic $p$ case. Can someone give a solution to that end?
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24 views

Prove a particular set is a ring with unity

I have to show that $(\mathbb{Z}[G],+,.)$ is a unitary ring, where $$\mathbb Z[G]=\{\sum_{g \in G} a_g.g| a_g \in \mathbb Z, a_g \neq 0, \text{only for finite g in G}\}$$ with $G$ group and $(\sum ...
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1answer
38 views

Problem on the number of generators

I have got stuck with two generator problems: The ideal $(zx,xy,yz)$ can't be generated by $2$ elements The ideal $(xz-y^2,yz-x^3,z^2-xy)$ can't be generated by $2$ elements Here the ...
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33 views

Orbit and stabaliser of $2\times2 * 2\times1$ matrices

I have the group action of matrix multiplication, meaning: $g((x,y))=\begin{pmatrix}a&0\\0&b\end{pmatrix}$$ \begin{pmatrix}x\\ y\end{pmatrix}$ ...
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37 views

Field Theory : What is wrong with this “homomorphism”?

Let $E/F$ be a field extension and $\alpha \in E$ be algebraic over $F$. Let $m(x) \in F[x]$ be irreducible and such that $m(\alpha) \neq 0$. Define $$ \phi : F(\alpha) \to F[x]/(m) $$ by ...
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3answers
71 views

Remainder of dividing $x^{137}+x+1$ by $x+5$

In $\mathbb{Z}_7[x]$, what is the remainder of dividing $x^{137}+x+1$ by $x+5$? I can not find how to solve this problem of modular arithmetic. Anybody could tell me only as I proceed to solve this ...
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0answers
48 views

Cardinality of ring having more than one left inverse for some element! [duplicate]

Suppose $R$ is a ring with unity $1$ and for some $a\in R$ there exists more than one left inverse of $a$ in $R$. Show that $R$ has infinitely many left inverses of $a$. I am trying to define a ...
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2answers
78 views

How do you maximize a polynomial over an integer domain?

I am working on maximizing the polynomial $f:\mathbb{R}^N \rightarrow \mathbb{R}$ $$f(v):=\prod_{i=1}^{N}( v_i+\alpha_i^2)$$ over integer $n$-partitions of $P$, $n\leq N$: $$\left\{v \left|\, v_i \in ...
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47 views

Classics on abstract algebra and real analysis

I am going through Apostol's calculus volume 1. What a wonderful creation from Apostol. Even I could not imagine that such a book introducing the basic concepts so informally but easy-to-understand ...
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22 views

Kernel of homomorphism $A[X] \to B$ between integral domains [duplicate]

Let $A \leq B$ be integral domains, where $A$ is integrally closed and $B/A$ is an integral ring extension. Let further $\varphi : A[X] \to B$ be some homomorphism of $A$-algebras. Is the kernel ...
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33 views

Show that for commuting projections we have $\mbox{im}(PQ) = \mbox{im}(P) \cap \mbox{im}(Q)$

Let $P, Q \in \mbox{Hom}(V,V)$ be projections on $V$, i.e. linear mappings such that $P^2 = P$ and $Q^2 = Q$. Show that if $PQ = QP$, then $$ \mbox{im}(PQ) = \mbox{im}(P) \cap \mbox{im}(Q). $$ One ...
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2answers
59 views

Cant understand some definitions of abstract algebra, can you help me please?

I was reading different definitions of vector space but still I have some dark places without meaning... I cant understand exactly what type of relation is defined between the vector space and the ...
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55 views

If $X$ generates $\Bbb{Q}$ then $X\setminus\{x\}$ also generates $\Bbb{Q}$ [duplicate]

If $X$ is a generator subset of $\Bbb{Q}$ then for $x\in X$, $X\setminus\{x\}$ also generates $\Bbb{Q}$. Clearly if I can express $x$ as a combination of the remaining generators we are done. ...
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1answer
41 views

A Commutative Ring Having a Unique Prime Ideal (Dummit and Foote, Prob 7.4.40(i))

I am trying to solve Problem 7.4.40 from Dummit and Foote, a part of which states: Let $R$ be a commutative ring with $1\neq 0$ such that $R$ has exactly one prime ideal. Then every ...
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1answer
51 views

Normal subgroup of a finite group

Let $G=\langle X_1,X_2\rangle$. Can we say that if $X_1$ or $X_2$ is a normal subgroup of $G$, then $G=X_1X_2$?
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1answer
24 views

A uniform module as an intersection

Let $R$ be a semiprime, nonsingular ring with finite Goldie dimension u.dim $R_R$. (Nonsingularity means here that $Z(R_R)=0$, where $Z(R_R)$ is the set of elements $x$ of $R$ with $ann(x)$ is ...
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1answer
26 views

Prove that $C′$ is a subgroup of $G$.

Let $G$ be a group. Let $C′=\{ a\in G:(ax)^2=(xa)^2 \forall x \in G\}$. Prove that $C′$ is a subgroup of $G$. I already proved that this is closed under multiplication, but I don't know how to prove ...
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1answer
35 views

what are the p-orbits in the decomposition of Σ into p-orbits.

At the bottom in the proof of slow 2, what's the meaning of "restrict the action of G on Σ to an...on Σ"? Since I can't understand that so I don't know what are the p-orbits in the decomposition of Σ ...
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1answer
41 views

Question concerning the meaning of an equality sign in a commutative diagram

$\require{AMScd}$ I have the following question: Let $\mathscr{C}$ be a category, $X,Y,Z\in Ob(\mathscr{C}), \ f\in Mor(X,Y),\ g\in Mor(Y,Z)$ and $h\in Mor(X,Z)$. Question: What does the ...
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1answer
31 views

Proof Unicity of Splitting Field

Context : Definition : We say $f(x) \in F[x]$ splits over the field extension $E/F$ if $$ f(x) = c (x-\alpha_1)\cdots(x-\alpha_n) $$ for some $c, \alpha_1, \ldots, \alpha_n \in E$. A splitting ...
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27 views

Reverse the twisting of modular form

It is known that the twisting of the Fourier expansion of a modular forms by a Dirichlet character produce a modular form. ...
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31 views

Linear equation over $\mathbb{Z}/n\mathbb{Z}$

For given $a,b\in \mathbb{Z}/n\mathbb{Z}$ is there a criterion which allows one to determine whether there exists $x\in \mathbb{Z}/n\mathbb{Z}$ with $ax=b$?
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67 views

Is orbit a group?

Let a group $G$ act on a set $S$, and let $s$ be an element in $S$. The identity of the orbit of $s$ is $s$ itself and if $a$, $b$ are in the orbit of $s$, then $a b = g_1 g_2 s$, where $g_1$, $g_2$ ...
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1answer
37 views

Rapid question on minimal normal subgroups

Why does $N$ finite group, $M\unlhd N$ imply that $M$ contains a minimal normal subgroup of $N$? If $M$ is itself minimal in $N$ we have finished. But if not, we know that there exists a minimal ...
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4answers
141 views

Is 1/2 less than 1?

Prove that $\frac 12 \lt 1$ We're in the reals and we have the Field Axioms and Order Axiom. I know that for $x \lt y$, $y - x \gt 0$. However I think I'd be assuming what I;m trying to prove if I ...
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1answer
59 views

Does there exist a UFD having only finitely many irreducibles?

Does there exist a UFD (which is not a field) having only finitely many irreducible elements? Definition of a UFD is: $R$ is an integral domain ($R$ is a commutative ring having unity and no ...