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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Prove linear independence and spans with linear maps

Suppose that $V,W$ are vector spaces over $\Bbb{F}$ and that $T : V → W$ is a linear transformation. (a) Suppose that $T$ is one-to-one, and that $\{v_1, · · · , v_n\}$ is linearly independent in $V$ ...
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36 views

Show that the alternating group $A_9$ has no subgroups of index 8?

So far, I believe it's a proof by contradiction. Suppose that $H \leq A_9$ with $[A_9 : H] = 8$.. $|H| = |A_9|*8$(which is a large number)? then would this involve the 3-cycles? Quite stumped. Thank ...
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100 views

Isomorphism of direct product of quotient rings

How do I show that with $n = p^aq^b$ with $p,q$ distinct primes and $a,b \geq 1$ that $$\mathbb Z/n\mathbb Z \cong (\mathbb Z/p^a\mathbb Z) \times(\mathbb Z/q^b\mathbb Z)?$$ I am told that Bezout's ...
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96 views

Is every well ordered commutative nontrivial ring with identity an well ordered integral domain?

$\mathbb Z$ is up to ring isomorphism the only well ordered domain, that is, $\mathbb Z$ is a integral domain and every nonempty subset of $\{n \in \mathbb Z: n\geq 0\}$ has a least element. But what ...
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automorphism group of direct product of groups

I was working on a problem in group theory, which asks about the automorphism group of a direct product of groups. Okay, so I know that if $G,H$ are two groups whose orders are relatively prime, then ...
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74 views

Notation Question in proof of Kronecker-Weber theorem

I am trying to understand this proof of the Kronecker-Weber theorem by Franz Lemmermeyer http://arxiv.org/pdf/1108.5671.pdf. The set-up is this: $K/\mathbb{Q}$ is a cyclic extension of prime degree $...
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34 views

Finding the number of matrices up to similarity with characteristic polynomial

I need to find the number of similar matrices with characteristic polynomial $(x-1)^4(x-2)^3(x-3)^2$ $(x-1)^4$ can have the following forms $\begin{pmatrix} 1 & 1 & 0 & 0\\ 0 & ...
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51 views

Sylow counting - classifying groups of order 15

let $G$ be a group of order $15$. this is the argument I was given: We have $15 = 3\times 5$ so we start with $p = 5$ We have by Sylow's theorem that $N_5 = 1 \mod 5$ so $N_5 = 1$ or $N_5 \geq 6$. ...
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60 views

Finding the inverse of $2+\sqrt{5}+2\sqrt{7}$

Finding the inverse of $2+\sqrt{5}+2\sqrt{7}$ in the field $\mathbb{Q}(\sqrt{5},\sqrt{7})$. I know that all the elements of $\mathbb{Q}(\sqrt{5},\sqrt{7})$ are of the form: $a+b\sqrt{5}+c\sqrt{7}+d\...
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1answer
57 views

Showing two field extensions are equal

Let a,b $\in$ $\mathbb Q$ with b nonzero. Show that $\mathbb Q$($\sqrt a$)=$\mathbb Q$($\sqrt b$) if and only if $\exists$ c $\in$ $\mathbb Q$ such that a=b$c^2$. I am confused on how it is possible ...
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66 views

Decompose a vector space into invariant subspaces?

Consider the following proposition: Suppose $V$ is a finite dimensional vector space over a field $F$, and $K/F$ is a finite Galois extension with Galois group $G$. If $V$ has a $(K,K)$ bimodule ...
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61 views

Subgroups of $ \mathbb{Z}_n$ (integers mod $n$) [closed]

Is $\langle 15 \rangle$ a subgroup of $ \mathbb{Z}_{18}$ (the integers mod $18$)? There is a theorem in my book that says for every divisor $k$ of $n$, $\langle n/k \rangle$ is a subgroup of $ \...
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56 views

Galois theory about automorphisms of the field of rational functions

Suppose That $F$ is a field and $G=Aut(F(x))$ is the group of field automorphisms of the field of rational functions $F(x)$ and fix $F$, and that $E\subset F(x)$ is the fixed field of G. please prove ...
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2answers
159 views

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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33 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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48 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 $p$...
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128 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 $v_p(n)...
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53 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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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}) = \displaystyle{\...
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1answer
56 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
103 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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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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45 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 $x=2^{\frac{1}{2}}+5^{\frac{1}{3}}$, $x-2^{\frac{1}{2}}=5^{\frac{1}{3}}$, $x^3-3*2^{\...
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1answer
34 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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1answer
65 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) + g(...
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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 ...
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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 GL_k(\mathbb{R}),\;...
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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
36 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 $[K:F]=8$, i....
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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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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 $R=...
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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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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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102 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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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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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 $\sqrt{2}$,$\sqrt[3]{5}\in\mathbb{Q}(\sqrt{2}+\sqrt[3]{5})$...
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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 $\mathbb{Z} \oplus \mathbb{Z}$,...
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4answers
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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 ...
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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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112 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
45 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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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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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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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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1answer
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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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58 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
49 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 ...
4
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1answer
134 views

Quadratic number field which is Euclidean but not norm Euclidean [closed]

I am looking for an example of a quadratic field $\mathbb Q[\sqrt d]$ , with $d \equiv 2 $ or $3\pmod 4$ , whose ring of integers is Euclidean but not norm Euclidean. Please help. Thanks in advance....
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1answer
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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.
2
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2answers
103 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 nonzero ...