Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, among other topics.

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1answer
20 views

Find number of inversions in the permutation $X$. Given $A$, $B$, $C$ and $AXB = C$.

$$ A = \begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 2 & 3 & 1 & 5 & 7 & 6 & 4 \end{pmatrix} \\ B = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 ...
0
votes
1answer
46 views

Injectivity of composed homomorphisms. [on hold]

I'm doing this exercise: Let $R$ be a commutative ring and $S$ be a subset of $R$, with $S$ multiplicatively closed (that is: $1 \in S$, $ab\in S$ if $a,b\in S$ , $0$ is not in $S$ and $S$ has not ...
0
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3answers
37 views

Factor this polynomial into linear factors with coefficients in $F = \mathbb{Q}(2^{1/3}, i\sqrt{3})$

The polynomial is this: $x^3 -2$ Okay, so first I can create my field extension. I can easily extend the field to $2^{1/3}$. And I know the elements of the extension of $\mathbb{Q}(2^{1/3})$ can be ...
0
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2answers
28 views

Group Theory: how to find subgroups

I am trying to address my weak points with group theory, and thought I could learn this through an example: Let $G = (\mathbb{Z}_4 \times\mathbb{Z}_6, +)$. Find $3$ subgroups of $G$ of size $12$. ...
1
vote
1answer
26 views

Exercise: splitting field, showing that it splits

I need help with this exercise: Let $\alpha$ be a zero of $x^3+x^2+1$ in $\mathbb{Z}_2$. Show that $x^3+x^2+1$ splits in $\mathbb{Z}_2(\alpha)$. [Hint: There are eight elements in ...
0
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0answers
28 views

Can there be two non-isomorphic sets of permutations with a one-to-one match between i's in S1 and k's in S2 (see description)?

In thinking about this question — where sets of $M$ permutations of length $N$ ($M<N$) are defined as "isomorphic" if one permutation function can be found that, when applied to each permutation in ...
0
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2answers
25 views

Alternating Group $A_n$ does not have proper subgroup of index less than n, where n>4.

A proof is to be given for this. So what i have thought is: Let us assume to the contrary, i.e. it does have a subgroup of order m (say) less than n. Then, since $A_n$ is simple for n>4 , by ...
2
votes
0answers
27 views

When can a group be made into a ring? How `little' of the ring structure must be specified?

Given a (topological) abelian group $G$ and a (bicontinuous) $G$-bilinear map $\mu: G \times G \to G$, clearly $G$ becomes a (topological) ring by specifying $$ x y := \mu(x, y) \quad \forall x, y \in ...
3
votes
2answers
48 views

about center of group rings $RG$ and $(R/I)G$

Let $I$ be an ideal of a ring $R$. It is mentioned in the book An Introduction to Group Rings (by Sehgal and Milies) that the canonical homomorphism $RG \rightarrow (R/I)G$ maps $Z(RG)$, center of ...
4
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2answers
198 views

Questions about p-adic numbers

I got two questions about $p$-adic numbers: I often read that the field $\mathbb Q_p$ is much different than the field $\mathbb R$. An element of $\mathbb Q_p$ is of the form ...
0
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0answers
30 views

The maps $\delta: V \to B \otimes V$ and $\delta': V \otimes V^* \to B$.

Let $B$ be a coalgebra and $V$ a vector space. Suppose that we have a coaction $\delta: V \to B \otimes V$. Is the map $\delta$ equivalent to a map $\delta': V \otimes V^* \to B$? Thank you very much. ...
8
votes
0answers
74 views

Classification of all subrings

Let $R$ be an integral domain whose underlying additive group is finitely generated free and whose field of fractions $K$ is a finite Galois extension of $\mathbb{Q}$. Is there a method of ...
0
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2answers
38 views

Cyclic groups and other groups

Ok So I know that a cyclic group is a group that is generated by a single element like $\large{(Z_n,+)}$. Now I was wondering that if every group has a generator , and I found that the answer is yes ...
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2answers
26 views

Topology of $\Bbb{Q}_p$

Let $a\in \Bbb{Q}_p$. Is $ a+p^x\Bbb{Z}_p$ an open set around $a$ in the topology of $\Bbb{Q}_p$. Here $x \in \Bbb{Z}$. Also I have another question. Is $\mathbb{Z}_p$ open in $\Bbb{Q}_p$?
0
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2answers
55 views

Prove $Inn(G)$ is a normal subgroup of $Aut(G)$ [closed]

I found documents explaining how to prove $Inn(G)$ is a normal subgroup of $Aut(G)$, but can we prove for $G$? Also, I have another question along the same line. How do we show $Aut(G)/Inn(G) \cong ...
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2answers
55 views

If $\dim_F F[A] <n$, then $A$ is not cyclic. [closed]

Suppose that $A \in M_n(F)$. And the minimal polynomial of $A$ is irreducible. S0 $F[A]$ is a field extension of $F$ . I have to show : 1) If $\dim_F F[A] <n$, then $A$ is not cyclic. 2) if ...
1
vote
1answer
24 views

Find the order of U(36) using Euler ϕ−function, also find the order of 5 in U(36)

I was wondering if anybody can look over my proof and see if I made any mistakes or if their is a simpilar way of solving this problem. Find the order of U(36) using Euler ϕ−function, also find the ...
0
votes
1answer
35 views

Prove the Galois Group is Isomorphic to $S_3$

Prove G=$Gal(\mathbb{Q}(\sqrt[3]{2},i\sqrt{3}) : \mathbb{Q})$ is isomorphic to $S_3$ I know that the G has 6 automorphims, and $S_3$ has order 3! then consider polynomial $x^3-2 = ...
0
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0answers
45 views

Evaluating composition of functors

Let $R$ be a ring and $S$ its $n \times n$ matrix ring. We consider the categories $_R Q$ and $_S Q$ of their respective left modules. We define a functor $F \colon _R M \to _S M$ by $$ F(M) = M^n $$ ...
0
votes
1answer
59 views

How to show $1+\sqrt 2$ generate an infinite cyclic group of units in $\mathbb Z[\sqrt 2]$?

The answers given here seem very convoluted: The units of $\mathbb Z[\sqrt{2}]$. Is it possible to provide a more explanatory proof?
1
vote
1answer
46 views

a=a^{-1} iff aa=e

Dr. Pinter's "A Book of Abstract Algebra" presents the following exercise from Chapter 4, "Elementary Properties of Groups." Let G be a group. Let a,b,c denote elements of G, and let e be the ...
0
votes
0answers
50 views

Singularities in the weighted projective space

Is there an explicit criterion for checking that a hypersurface $f=0$ of degree $d$ and in $\mathbb{P}(a_0,\ldots,a_n)$ is smooth ? I could not convince myself that the criterion $\nabla f\neq 0$ ...
3
votes
4answers
55 views

Linear independence of $\sin(x)$, $\sin(2x)$, $\sin(3x)$ in Map($\mathbb{R},\mathbb{R}$)

It seems rather obvious to me that $\sin(x)$, $\sin(2x)$, $\sin(3x)$ are linearly independent in $\operatorname{Map}(\mathbb{R},\mathbb{R})$, but I'm not sure how to prove it (or disprove it if I'm ...
0
votes
2answers
40 views

infinite order of element with element in an infinite group

If $G$ is a infinite group, then $G$ must have an element of infinite order. Is this true? I know that if $G$ is infinite cyclic, then it's isomorphic to $\mathbb Z$. (I guess fact is irrelevant ...
2
votes
1answer
32 views

Transitivity of the discriminant of number fields

Let $M/L/K$ be a tower of number fields with discriminant of $M/K: d_M$ and of $L/K: d_L$. I would like to find a transitivity theorem for the discriminant and by letting $p_i$ and $q_i$ be integral ...
0
votes
3answers
51 views

Question about field extentions?

if $\mathbb{Q}(\sqrt{3}) $ can be looked at as the field of rational numbers with $\sqrt{3}$ appended to it, and can be furthermore looked at like $\mathbb{Q}[x]/x^2 - 3$ what does a field extention ...
0
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0answers
26 views

Does the dual behave well with base change?

Let $R$ be a ring, $M$ an $R$-module and $R \to S$ a ring homomorphism. I wondered under which conditions we have an isomorphism $$ Hom_R(M,R) \otimes_R S \xrightarrow{\sim} Hom_S(M \otimes_R S, S). ...
4
votes
4answers
358 views

infinitely many ideals

does the ring $\Bbb Z_2[x]$ have infinitely many ideals like $\Bbb Z[x]$? How do you know if a ring has a finite number of ideal. particularly asking about seemingly large rings.
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0answers
12 views

Construction of lattice in a given genus

Recently I needed to find a lattice in a given genus (which I attained as an orthogonal sum of lower dimensional lattices), which led me to the question how one can costruct such a lattice in general. ...
0
votes
1answer
13 views

What are the benefits of using reduction map and lift instead of function and inverse image?

I'm reading William Stein's: Elementary Number Theory: Primes, Congruences, and Secrets. And I found this definition. It employs the concept of reduction map and lift, but it seems to be very ...
0
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2answers
38 views

In a ring, if addition is commutative, does it implies that multiplication is commutative?

This might be a silly question after all. But I guess that assuming associativity and commutativity for $+$ in a ring, one could reorder the operations and obtain the commutativity for $\cdot$.
0
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0answers
46 views

Countability of $\mathbb{Q}$ using two sequences. [closed]

I'm trying to prove the countability of $\mathbb{Q}$ using the following problem. For integers $n\ge 1$ , let $a_n=k$ if $3^k$ divides $n$ and $3^{k+1}$ doesn't divide $n$. Let $b_1=2$, and for ...
1
vote
1answer
58 views

Is $\mathbb{Q}(\sqrt{3})$ in someway related to Quotient ring?

I can't help but notice that they look exactly the same. For example: $\mathbb{Q}(\sqrt{3})$ = $\lbrace p + q\sqrt{3}:p,q \in \mathbb{Q}\rbrace$ That seems pretty much exactly an ideal. Only the ...
5
votes
2answers
49 views

Must a non-simple group have a normal Sylow subgroup?

In class, one way we're taught to prove a group is not simple is to exhibit a normal Sylow subgroup. I'm wondering if the converse is true, i.e. if a group is not simple, must it have a normal Sylow ...
0
votes
1answer
45 views

If $abc$ is its own inverse…

Dr. Pinter's A Book of Abstract Algebra presents the exercise: Let $G$ be a group. Let $a,b,c$ denote elements of $G$, and let $e$ be the neutral element of $G$. Prove the following: ...
1
vote
2answers
23 views

Finding the conjugates, why can they argue this way?(exercise)

In one exercise I am supposed to find the conjugate of $\sqrt{2}+i$ over $\mathbb{Q}$. I found the answer by finding irr$( \sqrt{2}+i,\mathbb{Q})$, and then solving the polynomial finding all the ...
5
votes
1answer
66 views

How general $ [X,[Y,Z]] \cong [X \times Y, Z] $ is?

In some algebraic categories (abelian groups, modules on a ring) and in pointed spaces categories the following holds: $$ [X,[Y,Z]] \cong [X \times Y, Z] $$ In wich generality this lemma holds?
4
votes
5answers
97 views

If $ab=ba$, Prove $a^2$ commutes with $b^2$

From Dr. Pinter's "A Book of Abstract Algebra": Given $a$ and $b$ are in $G$ and $ab=ba$, we say that $a$ and $b$ commute. Prove $a^2$ commutes with $b^2$ I tried: $$ ab=ba $$ $$ ...
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vote
1answer
46 views

Group Properties - “$a$” commutes “$b$”?

Dr. Pinter's "A Book of Abstract Algebra" presents this problem from Chapter 4: If $a$ and $b$ are in $G$ and $ab=ba$, we say that $a$ and $b$ commute. Assuming that $a$ and $b$ commute, prove the ...
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0answers
110 views
+50

Whether a functor is exact?

I am stuck with exercise $1$ of section $3$ of chapter $1$ in the book Cohomology of number fields by Neukirch. The exercise is to show that the functor from $A \rightarrow C^n(G,A)$ is exact, where ...
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4answers
48 views

How do I show that $b^8=a^3ba^{-3}$?

Suppose $G$ is a group. I am trying to show that for $a,b\in G$, if $aba^{-1}=b^2$, then $b^8=a^3ba^{-3}$. I am not even sure if this is true but I found this in Artin's Algebra. My work: ...
0
votes
1answer
22 views

substraction of groups in direct sum

Assume I have a sub group $G\leq \mathbb Z^n$ and I have $\mathbb Z^n = G\oplus \mathbb Z^m $ for some $m\leq n$. I want to deduce $G\cong\mathbb Z^{n-m}$. Is that true? how can I do it?
2
votes
2answers
41 views

Computing a certain $2014$-fold product using a particular associative binary operation $\ast$

$$x*y = 3xy - 3x - 3y + 4$$ We know that $*$ is associative and has neutral element, $e$. Find $$\frac{1}{1017}*\frac{2}{1017}*\cdots *\frac{2014}{1017}.$$ I did find that ...
0
votes
1answer
26 views

$\mathbb{Z}$-basis of quadratic ring

Definition. A quadratic ring $R$ is a commutative ring with $(R,+) \cong \mathbb{Z}^2$ (The additive abelian group of $R$ is isomorphic to $\mathbb{Z}^2)$ Lemma. If $R$ is a quadratic ring, ...
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votes
2answers
55 views

Elements in a field

Prove that there exists a field with $16$ elements. I know that there is a theorem that states that a finite field can only have $$ p^k $$ Where $p$ is a prime and $k$ is any positive integer. But ...
1
vote
1answer
46 views

Find image and kernel of $\varphi: \mathbb{Z}[x] \to \mathbb{C}$ given by $x \mapsto i$

Please help on this, I'm desperate: Consider the homomorphism $\varphi: \mathbb{Z}[x] \to \mathbb{C}$ given by $x \mapsto i$. Find: (a) the image; (b) the kernel; (c) exhibit the bijection of ...
-2
votes
3answers
58 views

Show that $\sqrt{5} \notin \mathbb{Q}(\sqrt{3})$

Show that $\sqrt{5} \notin \mathbb{Q}(\sqrt{3})$. I don't even know where to start. I can't find references to this in my textbook anywhere. I feel like the notation came out of nowhere.
0
votes
1answer
40 views

normal subgroup in $S_3$?

Is $\{(1),(1,3)\}$ a normal subgroup in $S_3$? I know that a normal subgroup means that the left cosets are equal to the right cosets.
0
votes
1answer
36 views

Why is $R=\{a+b(3)^{1/3}: a,b \in \mathbb{Q}\}$ not a ring? [duplicate]

I don't understand why $R=\{a+b(3)^{1/3}: a,b \in \mathbb{Q}\}$ is not a ring?
2
votes
3answers
83 views

About the group of units

I'm stuck at this section of the following problem: Let $R$ be a commutative ring and $S$ be a subset of $R$, with $S$ multiplicatively closed. Find the group of units of $S^{-1}R$. My try: ...