# Tagged Questions

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### Please check my proof on: $\sim$ is an equivalence relation $\Leftrightarrow S<G$

Problem: Let $\emptyset\ne S\subset G$, where $G$ is a group, and define a relation on $G$ by $a\sim b\Leftrightarrow ab^{-1}\in S$. Show that $\sim$ is an equivalence relation if and only if $S$ is a ...
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### Find intermediate fields of $\mathbb{Q}(\sqrt[3]{2}, \sqrt{3},i) \, | \, \mathbb{Q}(i)$

This is the problem I am facing: Compute the intermediate fields of the extension $K | \mathbb{Q}(i)$ where $K = \mathbb{Q}(\sqrt[3]{2}, \sqrt{3},i)$ and find the intermediate fields $M$ such ...
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### Galois closure of $\mathbb{C}(T,\sqrt{T^2+T+1})$ over $\mathbb{C}(T^3)$

I'm trying to solve the following problem. But it's too difficult for me. Let $\mathbb{C}(T)$ be a function field, and put $L:=\mathbb{C}(T,\sqrt{T^2 + T +1})$, $K:=\mathbb{C}(T^3)$. Let $M$ be a ...
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### How to show this is the minimal polynomial

I'm trying to the following problem. But I can't show some irreducibility of the polynomials. Put $L=\mathbb{C}(X,Y,Z)$, $\omega=\frac{-1+\sqrt{-3}}{2}$. Define two automorphism $\sigma, \tau$ of ...
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### Ring Structure: Definition

Let $R = \{a+bi\mid a,b \in \mathbb Z, i^2=-1\}$, with addition and multiplication defined by $(a+bi)+(c+di)=(a+c)+(b+d)i$ and $(a+bi)(c+di)=(ac-bd)+(bc+ad)i$, respectively. (a) Verify that $R$ is ...
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### Linear algebraic group [on hold]

Let A be a finite dimensional algebra over C . This means that there is a multiplication map $f : A \times A \to A$ that is bilinear ( it is not assumed to be associative). Define the automorphism ...
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### Is this $f:\mathbb{Z}_m^* \mapsto \mathbb{Z}_n^*$well defined?

This question is in regard to my previous one, I asked here: is this mapping well defined? I get that this mapping is well defined in $\mathbb{Z}_m$ and $\mathbb{Z}_n$ case but is not clearly so in ...
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### is this mapping well defined?

I have to prove (or disprove) that there exists a map $[a]\mapsto [at]$ from the set $\mathbb{Z}_m^*$ to $\mathbb{Z}_n^*$, if it is given that $m$ divides $n$, where $[a] \in \mathbb{Z}_m$ and ...
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### Existence of an element in a group of certain order if an element of other order exists

Show that if a group $G$ of order $1089=3^2\cdot 11^2$ contains an element of order $9$ then it also contains an element of order $33$. I tried to see what would Sylow theorems tell for this problem ...
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### Maximal ideal which isn't principal

Let $J=(x-2,x-y^2-3)$ ideal in the polynomial ring $\Bbb R[x,y]$. Please help me prove that $J$ is a maximal ideal which isn't principal, and that $\Bbb R[x,y]/J \cong \Bbb C$.
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### Ideals of $\Bbb Z/p^2q\Bbb Z$

Let $p,q$ be distinct primes. Then $\mathbb{Z}/p^2q\mathbb{Z}$ has 3 distinct ideals. $\mathbb{Z}/p^2q\mathbb{Z}$ has 3 distinct prime ideals. $\mathbb{Z}/p^2q\mathbb{Z}$ has 2 distinct prime ...
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### Existence of $p \times p$ matrices $A$ and $B$ over the field $\mathbb F_p$, $p$ prime, such that $AB-BA=I$. [duplicate]

Let $p$ be a prime number. Prove or disprove that there exists $p\times p$ matrices $A$ and $B$ over a field $\mathbb F_p$ with $AB-BA = I$. With the aid of MAPLE i was able to find out that ...
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### $x\in K(t)$ is algebraic over $K$ if and only if $x\in K$ (with $t$ transcendental over $K$)

I already proved the following: Lemma A: Let $K$ be a field. Let $t$ be transcendental over $K$. Let $f \in K[X] \backslash K$. Then $f(t)$ is transcendental over $K$. Proof: Because $t$ is ...
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### Are the rings $R=\mathbb{Z}[x]/(x^2+7)$ and $R'=\mathbb{Z}[x]/(2x^2+7)$ isomorphic?

I tried to use this method: Suppose there exists a $\phi$, that $\phi$ sends $2|_R$ to $2|_{R'}$. Then $$A=\mathbb{Z}[x]/(x^2+7)/(2), B=\mathbb{Z}[x]/(2x^2+7)/(2).$$ $\phi: A\rightarrow B$. It's easy ...
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### Total number of non isomorphic groups of order 122.

Let $G$ be group of order $122 = 61 \cdot 2 = p \cdot q$ , where $p < q$ are primes. I know that there exists a unique non abelian group of order $pq$ and one abelian non isomorphic group of order ...
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### Let $R$ a ring with maximum common divisor. If $a,b,c \in R$ such that $a|bc$ and $(a,b)=1$ then $a|c$.

Let $R$ a ring with maximum common divisor. Show that if $a,b,c \in R$ such that $a|bc$ and $(a,b)=1$ then $a|c$. Comments: I tried to use the Bezout's theorem, but in my course we saw it only ...
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### $M \oplus M \simeq N \oplus N$ then $M \simeq N.$

Let $M$ and $N$ be finitely generated $R$-modules where $R$ principal domain. Show that if $M \oplus M \simeq N \oplus N$ then $M \simeq N.$
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### Show that: $\frac{D_n}{\langle a\rangle}\simeq\mathbb{Z_2}$

Show that: $$\frac{D_n}{\langle a\rangle}\simeq\mathbb{Z_2}$$ where $D_n$ is dihedral group and $a$ is generator of order $n$.
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### Show that an integral domain $R$ is principal if and only if every submodule of a cyclic $R$-module is cyclic.

Good morning, I have difficulty with this problem: Show that an integral domain $R$ is principal if and only if every submodule a cyclic $R$-module is also cyclic.
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### Ring of linear transformations modulo finite rank transformations [closed]

Let $K$ be a field and $V$ be a vector space of countable dimension (infinite) over $K$, and let $L = L (V)$ be the vector space of $K$-linear transformations on $V$. Let $I$ be the ...
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### Finite dimensional vector space is finitely generated and a torsion module

The question is let $V$ be a finite dimensional real vector space and $T:V \rightarrow V$ be a linear transformation. Let $M={}_{\mathbb{R}[X]}V$ be the $\mathbb{R}[X]$-module defined in the usual way ...
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### Torsion-free but not free

The question is asking me to give an example of a finitely generated $R$-module that is torsion-free but not free. I remember in lecture, lecturer say something about the ideal $(2,X)$ in ...
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### help interpreting an abstract algebra test question

This is a take-home test problem, and I don't want help solving it, just understanding what it's asking. I've asked my prof a couple times, but she's either unwilling or unable to give me a straight ...
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### Find the order of a extension field.

I have the following exercice: Let $K$ be the two element field and $P(X)=X^3+X+1\in K[X]$. Show that $P$ is irreductible in $K[X]$. Let $\alpha$ be a root of $P$ in an extension of $K$. Show that ...
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This is a past exam question. Decompose each of the following elements as a product of irreducible: (a) $X^4+2 \in \mathbb{Z}_5[X]$ (b) $X^5+X \in \mathbb{Z}_2[X]$ (c) $X^5+4X^4-3X^3+X^2+7X+11 \in ... 1answer 29 views ### Irreducible in$\mathbb{Z}[X], \mathbb{Q}[X], \mathbb{R}[X]$and$\mathbb{C}[X]$Is the polynomial$2X^3-10X^2+50X+10$rrreducible in$\mathbb{Z}[X], \mathbb{Q}[X], \mathbb{R}[X]$and$\mathbb{C}[X]$. What I have done so far is, since this is a cubic function, therefore it will ... 1answer 36 views ### GCD of polynomials over$\mathbb{Z}_3f$and$g$are polynomials over field$\space \mathbb{Z}_3$.$f=X^4+X^3+X+2, \space g=X^4+2X^3+2X+2$. And I been asked to find the GCD of them. What I have done is using Euclidean algorithm. After ... 0answers 37 views ### Find a counterexample to the following statement:$\frac{M_1 \oplus M_2}{N} \simeq M_2$Suppose that$M$is a$R$-module such that$M = M_1 \oplus M_2$and let$N$submodule if$M$isomorphic the$M_1$. Find a counterexample to the following statement: $$\frac{M_1 \oplus M_2}{N} \simeq ... 1answer 25 views ### Show that if M is a R-module free, for all r \in R and m \in M such that r \neq 0 e m \neq 0 we have rm \neq 0. is it true that if M is a R-module free, for all r \in R and m \in M such that r \neq 0 e m \neq 0 we have rm \neq 0. Comments: I tried to do the following: Suppose that rm = ... 1answer 21 views ### Let R be a UFD and p,q,r \in R. pq=r^3 and \gcd(p,q)=1 then p,q are cubes up to associates. I'm not too sure how to prove this statement. This seems like a relatively small problem however I can't for the life of me figure out how to start this so I don't really have any working to show. I ... 2answers 38 views ### Minimal polynomial reducible modulo every prime p Suppose K = \mathbb{Q}(\alpha) with \alpha = a + b\sqrt{D_1}+c\sqrt{D_2}+d\sqrt{D_1D_2} with D_1,D_2 \in \mathbb{Z}. Prove that the minimal polynomial m_\alpha(x) for \alpha over ... 3answers 86 views ### This ideal is not maximal [duplicate] I'm trying to prove this ideal:$$(x^2+y^2+z^2+x+y+z,x^5+y^5+z^5+2(x+y+z),x^7+y^7+z^7+3(x+y+z))\subset \mathbb C[x,y,z]$$Can't be maximal. In order to do so, I'm using the Nullstellensatz ... 1answer 40 views ### \mathbb{C}/\mathbb{Z} \cong \mathbb{C}-{0} I have to prove that \mathbb{C}/ \mathbb{Z} \cong \mathbb{C}-\{0\} holds. I`m using this theorem: If \phi: \mathbb{C} \rightarrow \mathbb{C}-{0} is a homomorphism and H=Ker(\phi), with H as ... 1answer 62 views ### Show that f is diagonalizable Given an endomorphism f on the vector space on \mathbb{R} of dimension n such that f(f(x))=3f(x)-2x. Let E_1=\ker(f-Id) and E_2=\ker(f-2Id). Show that: 1.E_1 and E_2 form a direct ... 1answer 40 views ### Subgroups of (\mathbb Z_n,+) The problem is to define all subgroups of (\mathbb Z_n,+), n \in \mathbb N. My guess is if n is prime number, then there is only trivial subgroups. If n is not prime, then I can factorize it, and ... 3answers 52 views ### If f,g are two endomorphisms of E such that f(g(x))=g(f(x)) and g is nilpotent show that: f is invertible => f+g is invertible If f,g are two endomorphisms of E such that f(g(x))= g(f(x)) and g(x) is nilpotent show that: A) If f(x) is invertible then f+g is invertible too. B) If f(x)+g(x) is invertible then ... 2answers 48 views ### Field extensions and gcd Let L|K be a field extension and let u, v \in L be algebraic elements over K such that [K(u):K]=n and [K(v):K]=m. Show that if \gcd(m, n)=1 then Irr(v, k) is irreducible on K(u). ... 1answer 135 views ### Galois group of the quintic polynomial X^5+X+1 I'm trying to find the Galois group of the polynomial p(X)= X^5+X+1 over \mathbb Q. First, one notes that, if \omega is a primitive cubic root of unity, then it is a root of p(X). So, ... 1answer 53 views ### eH \in G/H is the only element with finite order \newcommand{\ord}{\text{ord}}This is a question in my book: G is an infinite Abelian group and H=\{ a \in G \mid \ord(a) < \infty \} is a normal subgroup of G, show that eH is the ... 2answers 61 views ### Minimal non solvable group is simple I suppose to prove in my homework that Every minimal non solvable group is simple. I can't find the way. Thank you. 1answer 113 views ### An ideal which is not maximal in \mathbb{C}[x,y,z] Show that$$J=(x^2+y^2+z^2+x+y+z, x^5+y^5+z^5+2(x+y+z), x^7+y^7+z^7+3(x+y+z))$$is not the maximal ideal$m=(x,y,z)$in$\mathbb{C}[x,y,z]$. 1answer 35 views ### If R and S are artinian and finite dimensional algebras respectively, then the tensor product of them is artinian. Let$R$be an artinian algebra and$S$be a finite dimensional algebra over the field$k$. How can i show that$R\otimes_kS$is artinian? I know that$S$is also artinian since it is finite ... 1answer 89 views ### How many normal subgroups does a non-abelian group$G$of order$ 21$How many normal subgroups does a non-abelian group$G$of order$21$have other than the identity subgroup$\{e\}$and$G$? 1) 0 2) 1 3) 3 4) 7 I think option 1 is incorrect because every group ... 2answers 32 views ### Subfields and Isomorphisms. Let$E$and$F$be subfields of a finite field$K$. Show that if$E$is isomorphic to$F$, then$E=F$. I am considering the subfield property and isomorphisim, is that ok? 0answers 65 views ### invariants of group action by algebra automorphism I am trying to prove the following statement, but I'm having a lot of trouble with it: Let$k$be an infinite field. Let$A$be a commutative$k$-algebra. Let$G$, a group, act on$A$by algebra ... 1answer 35 views ### Proves or counterexamples in retraction and coretractions of modules Any tip for proving or counterexamples that the following morphism of$\mathbb Z$-modules${\mathbb Z} \to {\mathbb Q}$is not a retraction and${\mathbb Q} \to {\mathbb Q/\mathbb Z} $is not a ... 1answer 41 views ### Representation of a subgroup I'm trying to solve the following problem. Suppose there is a$V$, representation of$G$, and a subgroup$H\leq G$with index$|G:H|=3$. Given that$V$seen as a representation of$H$is a direct sum ... 1answer 43 views ### Does the Cayley digraph$C$= [(12)(34),(123):$A_4$] have a Hamiltonian Circuit? This is a problem I'm working on for a friend of mine. I haven't been able to solve it, or make much progress. I have drawn the digraph, and it consists of four directed cycles of three vertices all ... 1answer 46 views ### Ideals on Rings?How do i Define them? How do I define all the possible ideals of a given Ring-Set? Example on$Z(m)$. Do I stop when I find enough ideals that their union give's me my given set?? 1answer 49 views ### Is there a specific method to finding a basis for vector spaces over$\mathbb{Q}\$ ?

I am stuck on the first one but there are 5 questions on this so I really need help with the process. If anyone can help with any of the following. i) Find a Basis for the field K = ...