# Tagged Questions

97 views

### Let $x = 2441921$. Factor $x$ into a product of primes.

Let $x = 2441921$. Factor $x$ into a product of primes. I found that: $1519^2 −x=−134560= −2^5 ·5 · 29^2$ and $1541^2 −x=−67240= −2^3 · 5 · 41^2$. I am trying to figure out what to do from here. ...
42 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 ...
38 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 ...
37 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 ...
80 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 ...
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 ...
51 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 ...
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 ...
84 views

### Linear algebraic group [closed]

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 ...
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 ...
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 ...
84 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 ...
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 $\Bbb R[x,y]/J \cong \Bbb C$.
65 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 ...
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 ...
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 ...
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 ...
202 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 ...
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 ...
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 $M \simeq N.$
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 $n$.
51 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.
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 ...
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 ...
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 ...
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 ...
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 ...