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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Show that $a(-1) = (-1)a = -a $.

In a ring $R$ with identity 1, show that $$a(-1) = (-1)a = -a \qquad\forall\, a \in R$$ I have started with $a + (-a) = 0$ but cant proceed from here.
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4answers
57 views

Prove the only homomorphism between groups with coprime orders is trivial. [on hold]

Deduce that if $\gcd(|G|, |H|)=1$ then the only homomorphism $\phi : G \rightarrow H$ is trivial.
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2answers
12 views

Prove the order of the group homomorphism of an element divides the order of the element.

Let $\phi : G \rightarrow H$ be a group homomorphism. Prove $\forall g \in G$, the order of $\phi(g)$ divides $g$. I've gotten to the point where I've shown that if, $ord(\phi(g)) < ord(g)$ then ...
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1answer
36 views

Sum of Two Squares in Ring Theory

Show that a prime $p$ in $\mathbb{Z}$ is a sum of two squares iff -1 is a square in $\mathbb {Z}_{p}$. This example belong to my ring theory book didnt have ideal. i read in number theory that If ...
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1answer
25 views

How many pairs of polynomials $(U,V)\in \Bbb Z[x]^2$ such that $P=U^2+V^2$ for a given polynomial with integer coefficients?

This question is no more than curiosity question. For integers we know that a positive integer $n$ is a sum of two squares if and only if for any prime $p$ such that $p\equiv 3 \mod 4$ we have ...
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0answers
16 views

Decomposition of an algebraic group into unipotent radical, component group and Levi subgroup

Consider the following statement : Given an algebraic group $G$ (over $\mathbb{R}$ or $\mathbb{C}$), there is a decomposition $$G = (G/G^o) \times \left( (G^o/R_u(G^o)) \ltimes R_u(G^o)\right)$$ ...
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1answer
47 views

Let $\phi:R[X] \rightarrow S[X]$ be a unital ring homomorphism. Prove if $f(x) \in R[X]$ is reducible, then $\phi(f(x))$ is reducible.

Let $R,S$ be integral domain and let $\phi: R \rightarrow S$ be a unital ring homomorphism. Define $\Phi:R[X] \rightarrow S[X]$ such that $\Phi(\displaystyle{\sum_{i=0}^m r_i X^i}) = ...
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1answer
49 views

A problem of field in abstract algebra

If $V$ is a finite-dimensional vector space over the field $K$, and if $F$ is a subfield of $K$ such that $[K:F]$ is finite, show that $V$ is a finite-dimensional vector space over $F$ and that ...
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1answer
47 views

Find all ring homomorphisms from $\Bbb Q$ to $\Bbb R$ [duplicate]

My question is find all homomorphism $ f: \Bbb Q \to \Bbb R$. I think I should use ring isomorphism theorem to do this problem, but I just don't know how to do this.
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0answers
40 views

Writing an abelian group as a direct sum of cyclic groups

I have this problem Determine an isomorphic direct sum of cyclic groups, where $V$ is an abelian group generated by $x,y,z$ and subject to:: $x+y=0, 2x=0, 4x+2z=0, 4x+2y+2z=0$ So I wrote ...
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0answers
43 views

A problem about an algebraic number [duplicate]

Show that $2^{\frac{1}{2}}+5^{\frac{1}{3}}$ is algebraic over $\mathbb{Q}$ of degree $6$. Can I just construct $x=2^{\frac{1}{2}}+5^{\frac{1}{3}}$, $x-2^{\frac{1}{2}}=5^{\frac{1}{3}}$, ...
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1answer
29 views

Prove that for any $u, v \in R \times R \times R$, $Ru + Rv \neq R \times R \times R$.

Let $R$ be an integral domain and $F$ be the field of fractions of $R$. Let $\phi: R \times R \rightarrow R \times R \times R$ be an R-module homomorphism.Prove that for any $u, v \in R \times R ...
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2answers
59 views

Prove that $x^3 + y^2$ is irreducible in $\mathbb{Q}[x,y]$.

Prove that $x^3 + y^2$ is irreducible in $\mathbb{Q}[x,y]$. My proof: $\mathbb{Q}[x,y] = \mathbb{Q}[x][y]$. Suppose $x^3 + y^2$ is reducible. Then $x^3 + y^2 = (y + g(x))(y + h(x)) = y^2(1 + h(x) + ...
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2answers
28 views

For $\mathbb{Q[x]}/ I \cong$ $\mathbb{Q}$, proving kernel

Let I $=<x-2>$. Prove $\mathbb{Q[x]}/I \cong \mathbb{Q}$ $\textbf{Pf:}$ Define $\phi: \mathbb{Q[x]} \rightarrow \mathbb{Q}$ by $\phi(f(x)) = f(2)$ I understand how to show that it is ...
4
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1answer
27 views

Representing Matrix Subgroups

Suppose I wanted to describe the subgroup of $GL_n(\mathbb{R})$ of matrices of the form $$ \left [ \begin{array}{cc} A & B \\ 0 & C \\ \end{array} \right ] $$ where $A \in ...
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0answers
23 views

When is the normality of field extensions transitive?

I want to answer the following question: Given field extensions $K \subset M \subset L $, when is the case that $M:K$ is normal and $L:M$ is normal implies $L:K$ is normal? In my algebra class we have ...
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1answer
15 views

Galois groups of extensions over $\mathbb{Q}$

Let $K=\mathbb{Q}(\sqrt[8]{2},i)$ and let $F=\mathbb{Q}(i)$. I need to show that $\operatorname{Gal}(K/F)\cong Z_8$ where $Z_8$ is the cyclic group of order 8. I have already shown that $[K:F]=8$, ...
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2answers
30 views

Classifying groups of order 4

Let $G$ be a group of order $4$ then either $ G \cong C_4$ or $G \cong C_2 \times C_2$ the proof my lecture gave goes as follows: Let $x \in G$ then by Lagrange $\text{ord}(x)$ divides $|G| =4 $ so ...
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1answer
43 views

Show that a variety is irreducible

How do I show that the variety $V = \{(x,y)\in k^2 \mid x-y=0\}$ is irreducible for an algebraically closed field $k$? One approach, I think, is to view $f(x) = x-y$ as an element in $R[x]$, where ...
3
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1answer
67 views

What does the notation $\mathbf{R}^\mathbf{R}$ mean?

I was reading the Princeton Review of GRE math subject test (4th edition), and one question was (page. 251) Example 6.24 Is the ring $\mathbf{R}^\mathbf{R}$ an integral domain? ...
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2answers
35 views

Is $\mathbb{Z}\times\mathbb{Z}/((6,5),(3,4))$ is finitely generated?

Let $A$ be the quotient of the free abelian group $\mathbb{Z}^2$ by the subgroup generated by $(6,5)$ and $(3,4)$. the question is $A$ is finitely generated? and if Yes. Can we Decompose it into a ...
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1answer
48 views

When is $\mathbb{Q}(x)$ a finite extension of $\mathbb{Q}$?

Let $x\in\mathbb{C}$, and consider the field $\mathbb{Q}(x)$. If this field is a finite extension of $\mathbb{Q}$, then $x$ is algebraic over $\mathbb{Q}$, so satisfies some polynomial $P$. Any field ...
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2answers
13 views

the cardinal of the set of left cosets the same as the cardinal of the set of right cosets?

is $|G/H| = |H/G|$ where $G/H$ is the set of left cosets of H in G, and $H/G$ the set of right cosets of H in G? I know that $|gH| = |H| = |Hg|$ but I don't see how $|G/H| = |H/G|$, even though ...
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6answers
119 views

Is $\mathbb{Q}(\sqrt{2},\sqrt[3]{5})=\mathbb{Q}(\sqrt{2}+\sqrt[3]{5})$?

I understand that $\mathbb{Q}(\sqrt{2}+\sqrt[3]{5})\subseteq\mathbb{Q}(\sqrt{2},\sqrt[3]{5})$ But I am struggling to algebraically show that ...
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0answers
30 views

What do we know about a group when we know it's order? [on hold]

Specifically, if a Group has order $rs$, what can we automatically know about this group?
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1answer
59 views

Determine a basis for $\mathbb{Z} \oplus \mathbb{Z}$ which determines a basis for the submodule $N$ generated by $(6,9)$

Proof Clearly the rank of $\mathbb{Z} \oplus \mathbb{Z}$ is $2$, so we must have that the rank of $(6,9)$ is $\leq 2$.Let $e_1 = (1,0)$ and $e_2 = (0,1)$ be a basis for $\mathbb{Z} \oplus ...
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4answers
64 views

Ring homomorphism from $M_3(\mathbb{R})$ into $\mathbb{R}$

I was working on this problem Find all ring homomorphisms from $M_3(\mathbb{R})$ into $\mathbb{R}$. My attempt:- I found that if we have any ring homomorphism $\phi$, then $\ker(\phi)$ should ...
2
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0answers
24 views

Examples for Burnside problem.

What are some examples for Burnside Problem- example of an infinite finitely generated torsion group - except Grigorchuk group. I have studies Grigorchuk group as an counterexample which was first ...
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1answer
38 views

Frobenius kernel is regular normal elementary abelian p-subgroup?

I'm attempting Exercise 3.4.6 in Dixon & Mortimer's book on Permutation Groups: Let $G$ be a finite primitive permutation group with abelian point stabilisers. Show that $G$ has a regular ...
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1answer
32 views

About the special linear group $SL(n,\,\mathbb{Z})$

I found the following relation: $$SL_n(\mathbb{Z})=\langle e+e_{12},\, e_{12}+\dots+e_{n-1, n}+(-1)^{n-1}e_{n1}\rangle$$ where $e_{ij}$ is the matrix with its $(i,\, j)$th entry $1$, and all other ...
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2answers
31 views

Definition of a Finite Normal Extension

I'm reading Fraleigh's A First Course in Abstract Algebra, Seventh Edition, and he makes the following definition on page 448: A finite extension $K$ of $F$ is a finite normal extension of $F$ if ...
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1answer
17 views

injectivity. identity map of a ring to its tensor product

Let $B$ be an $A$-algebra, $f: B \rightarrow B \otimes_{A} B$ is defined by $f(b)=b\otimes 1$. Is $f$ injective? I know the definition of tensor product and started from representing as $(b,1)=\sum ...
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2answers
15 views

Question about the centralizer and conjugacy classes

Let $G$ be a finite $p$-group and $H$ a non trivial normal subgroup of $G$. I want to prove that $H\cap Z(G)\neq 1$. I define a relation in $H$ by $x\sim y$ if and only if there exists $g\in G$ such ...
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3answers
71 views

find the minimal polynomial over $\mathbb{Q}$ of the algebraic number $\mathbb{(1+\sqrt{5})/2}$ [on hold]

find the minimal polynomial over $\mathbb{Q}$ of the algebraic number $\mathbb{(1+\sqrt{5})/2}$ . and write down the conjugates of the number over $\mathbb{Q}$
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1answer
34 views

Finding the irreducible polynomial of $\zeta_6, \zeta_8, \zeta_9$ over the field $\mathbb{Q}(\zeta_3).$

How can I get started on finding the irreducible polynomial of $\zeta_6, \zeta_8, \zeta_9$ over the field over $\mathbb{Q}(\zeta_3)?$ Should I construct field extensions and then use the degrees of ...
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0answers
24 views

the definition of Semigroups as S[x] [on hold]

how is the definition of S[x]? In rings it is said to be polynomial but in semigroup what is like that?
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0answers
29 views

problem about p-cycles [on hold]

Let a be a cycle in $S_n$ so that $a\neq(1)$ And $a^p=(1)$ with $n/2<p\leq n$ and $p$ being prime. Prove that $a$ is a p-cycle.
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43 views

Why $K = (X_1, X_2, …)$, the ideal generated by $X_1, X_2, …$ not finitely generated as a R-module?

Let $R = \mathbb{Z}[X_1, X_2, \dots]$ be the ring of polynomials in countably many variables over $\mathbb Z$. Why $K = (X_1, X_2, ...)$, the ideal generated by $X_1, X_2, ...$ is not finitely ...
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2answers
42 views

is 0 in the following Ideal?

Given $R=\mathbb R[x]$ and $I=(2x^3-3x^2+2x-3)+(2x^2-x-3)$ Is an Ideal of R? I don't understand what the quantity I is... Am I supposed to sum them together giving $2x^3-x^2+x-6$ Now here's the ...
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1answer
19 views

are calculations in ideals associative?

Say if we have $I+a+bX+cX^2+I+I$, can we rearrange the order to how we like? Because you can always imagine $0+I$ when the ideals are written consecutively.
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2answers
32 views

Nilpotent elements in the quotient ring of a polynomial ring

If $F$ is a field and $p(x) \in F[x]$, prove that the ring $R=F[x]/(p(x))$ has no nonzero nilpotent elements iff $p(x)$ is not divisible by the square of any polynomial. (==>) $R$ has no ...
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0answers
51 views

When this operation is associative?

I am looking for all possible positive values of $\alpha$ such that the binary operation (on natural numbers) defined by $$m\circ n = mn + \lfloor\alpha n\rfloor\lfloor\alpha m\rfloor$$ is ...
3
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1answer
43 views

Length of tensor product of finite length modules is finite

Let $R$ be a commutative ring. If $M$ and $N$ are finite length $R$-modules, then $M\otimes_R N$ has finite length, and $l(M\otimes_R N) \le l(M)l(N)$. I know the question has been posted ...
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1answer
36 views

About ring $( \frac{F[t]}{t^2})$

I was trying to study some properties of the group $GL_2 ( \frac{F[t]}{t^2})$ where $F$ is a field with $p$ elements. $p$ is a prime. I found that $ \frac{F[t]}{t^2}$ is a discrete Valuation ring. ...
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2answers
20 views

How to get an equivalent permutation [on hold]

What is the other form of the permutation $\sigma=(12)(345)$? how do I solve this kind of problem?
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1answer
38 views

show that $g(x)= x^n f(\frac 1x) \in \Bbb F[x]. $ where $\Bbb F $ is a field

Let $\Bbb F $ be a field and $f(x)=\sum_0^n a_i x^i \in \Bbb F[x]$. Show that $g(x)= x^n f(\frac 1x) \in \Bbb F[x]$ Show that if $r \neq 0$ is a root of $f(x)$ then $r^{-1}$ is a root of $g(x)$ Find ...
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1answer
40 views

I have to show $p=p(x-\lambda)$ if and only if they have the same zeros in $F$

Suppose $F$ is a field, $|F|\geq n \geq 2$. Given $\lambda \in F$ I know $p,p(x-\lambda)\in F[x]$ are irreducible monic polynomials. I have to show $p=p(x-\lambda)$ if and only if they have the same ...
0
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2answers
34 views

Show the two fundamental groups representation of the Klein bottle are the same

Show the two different representation of the fundamental groups of the Klein bottle $G=<a,b:aba^{-1} b>$ and $H=<c,d:c^2 d^{-2}>$. As we calculate in different and gett one result as ...
0
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2answers
30 views

Stabilizers of permutations

The problem is I need to list all elements in the stabilizer of the permutation $\sigma=(123)$ in $S_5$. I know that the stabilizer of a permutation $\sigma$ is the subgroup $$Stab_{{S_n}}(\sigma) = ...
0
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2answers
27 views

I have to show $p(x-\lambda)$ is an irreducible monic poynomial.

Suppose that $\mathbb{F}$ is a field , $|\mathbb{F}|\geq n \geq 2$. I know $p\in \mathbb{F}[x]$ is an irreducible polynomial and let $\lambda \in \mathbb{F}$ , I have to show $p(x-\lambda)$ is an ...