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 basis exists for a weird transformation

Suppose that $S : \Bbb{R}^2 → \Bbb{R}^2$ is a linear transformation such that $S^2 = S, S\ne 0 $ and $ S \ne I$ Prove that there is a basis for $\Bbb{R}^2$ with respect to which the matrix $A_S$ of ...
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0answers
37 views

$Z(G)$ acts on set of conjugacy classes by left multiplication

Let $z\in Z(G)$ then one can say that $(zx)^g=zx^g$. But it means that multiplication by $z$ create a bijection from conjugacy classes of $x$ to conjugacy classes of $xz$. Let ...
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0answers
17 views

Prove that a non empty finite subset H of a group G that shows closure is a subgroup.

This might be a little trivial but I am having difficulty in proving H will always consist of an identity element. Thanks
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0answers
29 views

Define Tame algebra

How would you define a tame algebra such that undergraduate students could understand it easily? Do someone knows a good book that has a nice way to explain it?
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3answers
35 views

Constructing the groupification of a semigroup (Vakil 1.5.G)?

In the first chapter of Ravi Vakil's Algebraic Geometry notes, he suggests a construction of the groupification $H(S)$ of an abelian semigroup $S$ by considering $S\times S/\sim$ where $(a,b)\sim ...
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0answers
20 views

I have to show Two monic polynomial of the same degree [closed]

I have to show Two monic polynomial of the same degree coincide if and only if they have the same zero set over algebraically closed field
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0answers
33 views

Determining Quotient group

Let $G$ be the group of linear functions under addition. Let $H$ be the subgroup of $G$ containing only the linear functions passing through the origin. How do I determine the group $G/H.$ What are ...
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1answer
11 views

Solving the Cubic Equation (using Lagrange Resolvents)

This is from my textbook. I am having trouble working out the calculations that the author skips over. So we start with the polynomial $\ X^3 - aX^2 + bX -c$ with zeros $x_1,x_2,x_3$. Then we define ...
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0answers
10 views

size of centralizer

Can someone tell me why the size of the centralizer of the semidirect product of $\mathbb{Z}_4\times \mathbb{Z}_4$ and $S_2$ is $4^2 \times 2 = 32$? I know the part why $4^2$ is multiplied by 2 that ...
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2answers
20 views

Checking for the reducibility of a polynomial using rational root theorem

When checking for the reducibility of a polynomial over $\mathbb{Z}$. I can either use the eisenstein criteria or contradiction. However, I am wondering if it is possible to use rational root theorem. ...
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1answer
28 views

Proof of that in an integral domain, every prime element is irreducible.

I would like to prove that in an integral domain $R$, every prime element $p$ is irreducible. I understand the case where $p = ab$ but the textbooks I have read do not address the case where $p \neq ...
3
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1answer
33 views

let $G=\mathbb{Q}^*$ and $\varphi: G \to G$ where $\varphi$ interchanges 2,3 in the prime power factorization. Prove it is a group isomorphism

let $(G, \cdot)=(\mathbb{Q}^x, \cdot) = (\{\frac{p}{q}\mid\frac{p}{q} \neq 0\}, \cdot)$ and $\varphi: G \to G$ where $\varphi$ interchanges 2,3 in the prime power factorization and $\varphi$ is ...
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0answers
17 views

Proving existence of Hermitian Adjoint in unusual way

For a map $T:V\rightarrow V$, we define the Hermitian adjoint to be the unique $T^*:V\rightarrow V$ such that $\langle Tu,v\rangle = \langle u, T^*v\rangle$. There are two things I'm required to ...
2
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1answer
40 views

Let $I = \{a+ ib \in \Bbb Z[i] : 2 \mid a-b\}$ then $I$ is a maximal ideal of $ \Bbb Z[i]$.

Let $I = \{a+ ib \in \Bbb Z[i] : 2 \mid a-b\}$ then $I$ is a maximal ideal of $ \Bbb Z[i]$. We consider an ideal $J$ such that $I \subset J\subset\Bbb Z[i] $. So there exists an element $p \in J$ but ...
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1answer
25 views

Prove vector spaces are isomorphisms

Suppose that $V$ is a finite-dimensional vector space over $\Bbb{F}$ and that $T : V → V$ is an isomorphism. Prove that if $S : V → V$ is also a linear transformation and $ST$ is an isomorphism, then ...
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2answers
30 views

Suppose $E$ is the quotient field of $D$ then find the relation between $D[x]$ and $E[x]$.

Let $D$ be an integral domain, then $D[x]$ is an integral domain and find its quotient field. Suppose $E$ is the quotient field of $D$. Then find the relation between $D[x]$ and $E[x]$. I have ...
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1answer
25 views

Let N be a normal subgroup of a group G. Prove that G is a p-subgroup if both N and G/N are p-subgroups. [closed]

Could someone please hint the outline of the proof of this Sylow p-theorem question? If N and G/N are p-subgroups, how do I "connect" this to G? Thanks in advance.
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1answer
20 views

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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1answer
32 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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0answers
34 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 ...
3
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1answer
54 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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1answer
45 views

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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0answers
29 views

Proof theorem of Lie's Algebra [closed]

I need help with a proof this theorems, anyone could be help? I need a proof this theorems. Corollary of Lie's Theorem: Let $L$ be a solvable subalgebra of $\text{gl}(V)$, $\dim V = n$ (finite). ...
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0answers
37 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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0answers
22 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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2answers
19 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 $N_5 \geq 6$. ...
3
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2answers
56 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: ...
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1answer
30 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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0answers
26 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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1answer
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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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0answers
28 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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1answer
26 views

How do I find this basis given matrix representations?

Here is the question: Consider the multiplication operator $L_A:{\mathbb R}^2\to {\mathbb R}^2$ defined by $L_A(x)=Ax$ where $A=\left[\begin{array}{cc}2 &0\cr1 &-1\end{array}\right]$. Find an ...
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2answers
57 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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4answers
57 views

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

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 ...
2
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1answer
27 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
49 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
49 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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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
28 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
24 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
31 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 ...