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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A subset of a polynomial ring and its ideal. [duplicate]

Let $P=K[x_1,\dots,x_n]$ be a polynomial ring over a field $K$ and $I = (f)$ be a principal ideal in $P$ generated by $f \in P - \{0 \}$. Moreover let $L \subset \{x_1, \dots, x_n \}$ and $\hat{P} ...
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40 views

Help me with this Group Question [on hold]

Let Sn := Bij({1, . . . , n}) the symmetric group with n elements equipped with composition of functions. (i) Show that Sn is a group. (ii) Let (e1, . . . , en) be the canonical base of Rn and Mn(R) ...
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1answer
49 views

Find the greatest common divisor of pairs of polynomials

I'm trying to find the greatest common divisor of $$p(x)=7x^3+6x^2-8x+4$$ and $$q(x)=x^3+x-2$$ where both $p(x),q(x)\in\mathbb{Q}[x].$ And if $d(x)=gcd(p(x),q(x)),$ I need to find two polynomials ...
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35 views

When does normal maximal subgroup have prime index?

Given finite group $G$, a normal maximal subgroup $H$, when is $[G:H]$ a prime? If $G$ is nilpotent, then the statement is true. But I am not sure about other $G$. Is there any counter-example ...
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28 views

$K$-affine $Hom$ functor relating to Polynomial Maps (Exercise in _Algebra: Chapter 0_ by Aluffi)

I'm currently learning some category theory and algebraic geometry, the basics at least. In Algebra: Chapter 0 by Aluffi, there is an exercise describing the relationship between morphisms of affine ...
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23 views

$f\in L[x,y]$ such that $f(x,0)=0$ implies $f=y g$ with $g\in L[x,y]$?

Suppose $L$ is an infinite field (or even algebraically closed; I'm not sure if it is necessary to add that hypothesis). If we have a polynomial $f(x,y)\in L[x,y]$ and $f(x,0)\equiv 0$, does that ...
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1answer
28 views

Proving group and Morphisms of groups [on hold]

Let Sn := Bij({1, . . . , n}) the symmetric group with n elements equipped with composition of functions. (i) Show that Sn is a group. (ii) Let (e1, . . . , en) be the canonical base of Rn and Mn(R) ...
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35 views

If $|F| = 8$ and $K$ is a subfield, show that $K = F$ or $K = \{0, 1\}$.

Let $K$ be a subring of a field $F$. If $|F| = 8$ and $K$ is a subfield, show that $K = F$ or $K = \{0, 1\}$. [Hint: Lagrange]. Lagrange's Theorem: If $H$ is a subgroup of a finite group $G$ I Then ...
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39 views

The center of a group

prove that for any group G, Z(G)=$\bigcap_{x\in G} C_{G}(\{x\})$ . In addition, show that if H$\subset$G , then $C_{G}(H)=\bigcap_{x\in H} C_{G}(\{x\})$ Z(G) is the center of a group $C_{G}(\{x\})$ ...
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1answer
55 views

If the tensor product of algebras $A \otimes B$ is unital, both $A$ and $B$ must be unital

It is clear that if $A$ and $B$ are unital algebras (over a field), then the tensor product $A \otimes B$ is also unital, with the unit being $1_A \otimes 1_B$. I came across an exercise that ...
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1answer
56 views

Kernel of a map in Graph Theory (toric ideals)

If we have an $n$-cycle with edges $e_1 =\{x_1,x_2 \}, e_2 = \{x_2, x_3 \},\dots, e_n = \{x_n,x_1\}$ with a $K-$algebra homomorphism $\phi: k[e_1,\dots, e_n] \to k[x_1,\dots, x_n]$ defined by ...
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26 views

Sylow counting to show group is isomorphic to semidirect product

Let G be a group of order $|G|=pq^m$ where $p, q$ are primes with $q^m<p$. Use a Sylow counting argument to show that $G\cong C_p \rtimes_h Q$ where $Q$ is a group with $|Q|=q^m$ and ...
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2answers
30 views

Show that a ring $R$ is a division ring if and only if, for each nonzero $a\in R$, there is a unique element $b\in R$ such that $aba = a$. [duplicate]

Show that a ring $R$ is a division ring if and only if, for each nonzero $a\in R$, there is a unique element $b\in R$ such that $aba = a$. $\Rightarrow$ Assume $R$ is a division ring. Let ...
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41 views

How to prove absence of a total order relation?

Show that on $\mathbb{C}$ (complex) there is no total order relation $≤$ such that both if the following properties hold $∀ (x, y, z) ∈ \mathbb{C}^3$, $x ≤ y \implies x + z ≤ y + z$ and $z ≥ 0, x ≤ ...
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30 views

How to show that a set of elements is a basis for the ring of integers of a number field?

Let $K$ be a number field of degree $n$ (that is $[K:\mathbb{Q}]=n$) with ring of integers $\mathcal{O}_K$. I know that there exists algorithms to find $\mathcal{O}_K$ and hence determine a ...
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1answer
36 views

Sylow's theorem for group of $2$ by $2$ matrices of determinant $1$ over the field of order $3$

Let $G=SL(2,\mathbb{F_3})$ - group of $2$ by $2$ matrices of determinant $1$ over the field of order $3$. (a) Find the order of $G$. I think it is $24$ but not sure how to verify it. (b) ...
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1answer
13 views

showing a function is surjective for isomorphisms

Consider a problem like the following. Prove that $ \mathbb{R} $ is isomorphic to the ring S of all $ 2 \times 2 $ matrices of the form $ \begin{bmatrix}a & 0\\0 & a\end{bmatrix}$, with $a ...
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40 views

Quick way to classify groups of small order

This is a past Qualifying exam on Algerbra : I'm curious if there is a quick way to solve Prob 1. To me, directly classifying groups takes too much time and I think I could not handle this in time ...
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1answer
70 views

Prove that these two fields are isomorphic.

I want to prove that $\bar{K}[V]/M_p \simeq \bar{K}$ where $K$ is a field, $\bar{K}$ is its algebraic closure and $$\bar{K}[V]=\bar{K}[x_1,...,x_n]/I_V,$$ where $I_V$ is the ideal attached to a ...
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22 views

Cosets of submodule

Suppose we have a ring $R$ and an ideal $J \subseteq R$. Let $M$ be an $R$-module and $N$ be a submodule of $M$. Assume for two elements $m$ and $m'$ of $M$ we have $$m+ JM=m'+JM.$$ Since $N ...
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31 views

Representation of a group and its quotient

Let $G$ be a (finite) group and let $N$ be a normal subgroup of $G$. Suppose that we have a representation $(V,\rho)$ of $N$ and a representation of $(V, \tau)$ of the quotient group $G/N$. Here $V$ ...
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72 views

Let $G$ be a group of order $35$. Prove that $G \cong C_5 \times C_7$.

Let $G$ be a group of order $35$. Prove that $G \cong C_5 \times C_7$. $G$ has subgroups of orders $5$ and $7$ by Lagrange's theorem? If so, call them $A$ and $B$. I know their intersection is ...
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23 views

Order of group $G = \{A\in M_2(\mathbb{Z}_p): \mathrm{det}A= \pm 1 \}$

Also, $p>2$ is a prime number. Firstly, it's obvious that $G \leq GL_2(\mathbb{Z}_p)$, and we know that $|GL_2(\mathbb{Z}_p)|=p(p^2-1)(p-1)$. Next, we define the homomorphic map ...
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70 views

Mathematics and cinema

I wander if anyone of you have some knowledge about relations between abstract algebra and cinema. I'm not searching for movies about mathematics or algebra; I'm searching for some kind of application ...
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27 views

Using a Sylow Counting Argument

Let G be a group of order $$|G|=pq^m$$ where $p$ and $q$ are primes and $q^m<p$. Show that $$G\cong C_p \rtimes_h Q$$ where $Q$ is a group with $|Q|=q^m$ and $h:Q\rightarrow Aut(C_p)$ is a ...
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22 views

Cartesian product to direct sum

I have no idea, how to prove rigorously the corollary from the proposition. I know that i can use the isomorphism $\phi:x_1e_1+...+x_me_m \in \oplus_i^mvect(e_i)\to (x_1e_1,...,x_m e_m) \in ...
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1answer
51 views

Show that no ring containing R can contain a root of g(x) = 3x +1

Show that if $R = \mathbb Z_6$ and $g(x) = 3x + 1 ∈ R[x]$, then $R[x]/(g(x)R[x])$ does not contain a root of $g(x)$. More generally, show that no ring containing $R$ can contain a root of $g(x)$. ...
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32 views

Field characteristic for a finite product of fields of characteristic $0$

Kind of a silly question, but is a finite product of fields of characteristic $0$ also of characteristic $0$? For instance, $\mathbb{C}$ has characteristic $0$, but then does $\mathbb{C}^n, n>1$ ...
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47 views

Ring of matrices has no nontrivial ideals [duplicate]

It is a theorem that a commutative ring is a field if and only if it has no nontrivial ideals. Clearly this does not hold in the noncommutative case. I am trying to show for instance that the ring of ...
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Reducibility over $Q$ implies reducibility over $Z$.

Let $f(x) \in$ $Z[x]$ , if $f(x)$ is reducible over $Q$ , then it is reducible over $Z$. I went through the proof from the book I'm reading , which starts as follows : We're given $f(x)$ is ...
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1answer
35 views

Two non-isomorphic groups which are epimorphic images of each other

Let $G$ and $H$ be two groups, I am looking for an example such that $G$ is an epimorphic image of $H$ and $H$ is an epimorphic image of $G$ (i.e. they are both quotients of the other group), but they ...
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For which rings does a polynomial in $R$ have finitely many roots?

Which infinite rings satisfy the following Every non-zero polynomial in $R[X]$ has only finitely many roots ? Are there such rings which are not integral domains ?
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1answer
22 views

Group of exponent $2$.

When I have a group $G$ of exponent $2$ and I know that all elements in $G-\{e\}$ are conjugated, is it right that $G$ is of order $2$? My try: For $g,h \in G - \{e\}$ the conjugation assumption ...
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1answer
23 views

A $R$-module over a ring $R$ with identity is free iff it is a direct sum of copies of $R_R$, where $R_R$ is $R$ considered as a module over itself

Let $R$ be a ring with identity and $M$ be an $R$-right module. Then $M$ is called free over $X \subseteq M$ if for every module $N$ with mapping $\alpha : X \to N$ we can extend it uniquely to a ...
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45 views

Group of finite order where every element has infinite order

An often given example of a group of infinite order where every element has infinite order is the group $\dfrac{\mathbb{(Q, +)}}{(\mathbb{Z, +})}$. But I don't see why every element necessarily has ...
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21 views

Subgroup of square generators. [duplicate]

Let $N$ be a subgroup of $G$ with $N$ being generated by $\{x^2|x \in G\}$. Prove that $N$ is a normal subgroup of $G$. And that $[G, G] \subset N$. My idea was to look at $G/N$ and take $f:G ...
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What's an Isomorphism?

I'm familiar with the definition (inverses and bijections, preserving operations) in the context of groups and vector spaces, the hoeomorphism of topological spaces, and have some feeling for the ...
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1answer
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Is/when is this property about the totative subset of $(\mathbb{Z}_n , +_n)$ true?

Is it possible to show (or when is it true); that for the group $\mathbb{Z}^+_n :=(\mathbb{Z}_n , +_n) $, there exists an $a \in \mathbb{Z}^+_n$ for each $z \in \mathbb{Z}^+_n$, where both $z+_n a$ ...
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Criterion for isomorphism of two groups given by generators and relations

When are two presentations of groups are isomorphic? In this post it is said: [...] find a set of generators of the first group that satisfies the relations of the second group [...] But I doubt ...
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Compact abelian group and Continuous functions

If $G$ is a compact abelian group, $\widehat{G}$ is the dual group of $G$,i.e. all the continuous homomorphism from $G$ to $S^1$,$S^1=\{z\in \mathbb{C}\big | |z|=1\}$. Show that the linear span of ...
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1answer
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Centralizer, normalizer, and center of a dihedral group

Let $A := \{1, r, r^2,..., r^{n-1}\}$. Compute $C_{D_{2n}}(A), N_{D_{2n}}(A),$ and $Z(D_{2n})$. So far I figured that all of the rotations are in the centralizer/normalizer, because all rotations ...
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Set of all permutations on n generating function [duplicate]

Show that $S_n = \langle (1\ 2), (1\ 2\ \ldots\ n) \rangle$ for all $n \geq 2$. I'm not sure how to approach this one.
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Generators for a free submodule of a free module

Suppose we are given a finitely generated free module $M$ over a ring $R$. Assume $N \subseteq M$ is a free submodule of $M$. If $B$ is a basis for $M$, does it follow that there exists a subset $A ...
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1answer
38 views

Prove that $\mu:G\times G \rightarrow G$ is a homomorphism if and only if $G$ is abelian.

Given $\mu:G\times G$ be the operation on a group $G$; that is, $\mu (a,b)=ab$. Prove that $\mu$ is a homomorphism if and only if $G$ is abelian. I have no problem on proving the necessary ...
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1answer
29 views

Show that $m\mathbb{Z}/md\mathbb{Z} \cong \mathbb{Z}/d\mathbb{Z}$ if and only if $(d,m)=1$

Show that $m\mathbb{Z}/md\mathbb{Z} \cong \mathbb{Z}/d\mathbb{Z}$ as rings without identity if and only if $(d,m)=1$. I know that if $\phi : A \to B$ is a epimorphism ring and $A$ is a unit ...
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1answer
24 views

Compute the matrix with the standard basis.

Let $X$ be the left representation of $S_3$. Compute the matrix $X(132)$ in the standard basis $S = {\{123,132,213,231,312,321}\}$. attempt: In general if $G = S_3 = {\{\pi_1, \pi_2,..,\pi_6}\}$, ...
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4answers
37 views

Show that $\ker(\phi)$ is a maximal ideal if and only if $B$ is a field

Let $A$ and $B$ two commutative rings with unity $1_A \not= 0_A$ and $1_B \not= 0_B$. Consider $\phi : A \to B$ a ring epimorphism. Show that if $\ker(\phi)$ is a maximal ideal, $B$ is a field. I ...
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1answer
17 views

Let $n \geq 2$ be an integer, and consider the group $Z_n:=(\{0,1,. . .,n-1\}, +_n)$. Let $k \in Z_n$ \ $\{0\}$.

Let $n \geq 2$ be an integer, and consider the group $Z_n:=(\{0,1,. . .,n-1\}, +_n)$. Let $k \in Z_n$ \ $\{0\}$. Show that the following statements are equivalent: (a) $\gcd(n,k)=1$, (b) the only ...
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19 views

Simplifying a coset

Let G be a group and let $M,N \leq G$ be normal such that $G = MN$. Prove that $G/(M \cap N) \cong (G/M) \times (G/N)$ I have found a solution to this question here: ...
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1answer
30 views

First isomorphism theorem application

Let G be a group with, $N\subset G$ a normal subgroup, And assume that $H$ is a subgroup of $G$, $H\subset G$. Further $HN=G$ and $H\cap N = \{e\}$ . Prove that $H$ generates the cosets of $N$ in ...