2
votes
4answers
36 views

Prove a property about the centralisator

Let G be a group and $U \subseteq G$ a subgroup. Let $x \in G$ be arbitrary. How to show that $C_G(xUx^{-1})=xC_G(U)x^{-1}$ where $C_G(U):=\{g\in G : gu=ug$ $\forall u\in U\}$ For the first ...
2
votes
2answers
37 views

If G is a group and N is normal in G with index d, then $x^d \in N$

I want to show the statement in the title. If $G$ is a group and $N$ is normal in $G$ with $[G:N]=d$, then $x^d \in N$ for all $x \in G$ I want to consider the image $xN$ of $x$ in $G/N$ $G/N$ has ...
2
votes
1answer
34 views

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 ...
4
votes
1answer
79 views

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 ...
3
votes
1answer
69 views

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 ...
1
vote
1answer
50 views

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 ...
2
votes
2answers
53 views

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 ...
0
votes
1answer
84 views

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 ...
0
votes
0answers
31 views

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 ...
0
votes
1answer
56 views

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 ...
6
votes
2answers
83 views

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 ...
0
votes
2answers
50 views

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 $\Bbb R[x,y]/J \cong \Bbb C$.
1
vote
0answers
64 views

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 ...
6
votes
1answer
98 views

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 ...
2
votes
2answers
45 views

$x\in K(t)$ 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 ...
0
votes
3answers
104 views

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 ...
0
votes
2answers
196 views

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 ...
2
votes
1answer
29 views

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 ...
7
votes
2answers
143 views

$M \oplus M \simeq N \oplus N$ 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 $M \simeq N.$
5
votes
3answers
50 views

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 $n$.
0
votes
2answers
50 views

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.
-1
votes
1answer
42 views

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 ...
1
vote
1answer
25 views

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 ...
2
votes
2answers
83 views

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 ...
2
votes
2answers
124 views

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 ...
2
votes
1answer
23 views

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 ...
2
votes
1answer
41 views

Irreducible factor decomposition

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 ...
0
votes
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 ...
1
vote
1answer
36 views

GCD of polynomials over $\mathbb{Z}_3$

$f$ 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 ...
0
votes
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 ...
1
vote
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 = ...
0
votes
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 ...
2
votes
2answers
39 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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
2answers
49 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 $K(u)$. ...
9
votes
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, ...
3
votes
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 ...
0
votes
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.
-4
votes
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 $\mathbb{C}[x,y,z]$.
2
votes
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 ...
2
votes
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 ...
1
vote
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?
3
votes
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 ...
0
votes
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 ...
3
votes
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 ...
0
votes
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 ...