Articles of abstract algebra

Show that, if $ a + bi$ is prime in $\mathbb{Z} $, then $a – bi$ is prime in $\mathbb{Z}$

Show that, if $ a + bi$ is prime in $\mathbb{Z} [i]$, then $a – bi $ is prime in $\mathbb{Z}[i]$ Since $\mathbb{Z}[i]$ is $ED$, then if $a+bi$ is irreducible then $a+bi$ is prime. But now how I can relate $a+bi$ with $a-bi$? Thanks for help.

Solve $f^n(x)=4^nx+\frac{4^n-1}{3}$ for $n$.

Solve $y=4^nx+\frac{4^n-1}{3}$ for $n$, where $n\in\mathbb{N_{\geq0}},$ $y\in2\mathbb{N}_{\geq0}+1$ and $x\in2\mathbb{N}_{\geq0}+1\setminus(4\mathbb{N}_{\geq0}+1\setminus8\mathbb{N}_{\geq0}+1)$. In clearer language the process is to start with some odd integer and subtract one and divide by $4$ repeatedly until you hit some number which would take you out of the odd integers if you continued further. The question asks you to find for any […]

Isn't $x^2+1 $ irreducible in $\mathbb Z$, then why is $\langle x^2+1 \rangle$ not a maximal ideal in $\mathbb Z?$

Isn’t $ x^2+1$ irreducible in $\mathbb Z$, then why is $\langle x^2+1 \rangle$ not a maximal ideal in $\mathbb Z[x]$? $ x^2+1$ cannot be broken down further non trivially in $\mathbb Z[x]$. hence, it’s irreducible in $\mathbb Z[x]$. Hence, shouldn’t $\mathbb Z[x]/\langle x^2+1 \rangle$ be a field and hence, $ x^2+1$ a maximal ideal in […]

How to build a subgroup $H\leq S_4$ having order $8$?

Can anyone explain me what would be the procedure for building a subgroup $H\leq S_4$ of order $8$? I started obviously as $H=\{id$. Then I added two disjoint $2$-cycles $(1\ 2), (3\ 4)$ for they commute and they are equal to their inverse, that is, $$H=\{id, (1\ 2), (3\ 4), $$ then I added the […]

Transcendence degree of $K$

Let $K$ be field. How do I proof that transcendence degree of $K[X_1,X_2,\ldots,X_n]$ is $n$? The set $\{X_1,X_2,\ldots,X_n\}$ is algebraically independent over $K$. So, I have to show that every subset of size greater than $n$ is algebraically dependent.

Is $x^6 + 3x^3 -2$ irreducible over $\mathbb Q$?

It seems that it is irreducible over $\mathbb Q$, I tried to apply Eisenstein’s criterion to $f(x+a)$ for some $a$, but it didn’t work. Also, I found a root $\alpha = \sqrt[3]{\frac{\sqrt{17}-3}{2}}$ and tried to prove that the degree of $\alpha$ over $\mathbb Q$ is 6, but I stuck in computation. Is there any approach […]

Fertile fields for roots of unity

Is there a field $K$, an odd prime $p$, and a positive integer $n$, such that $K[ζ] = K[ζ^p]$ where $ζ = ζ_{p^n}$ is a primitive $p^n$th root of unity not contained in $K$? In other words, can a base field $K$ be chosen such that adjoining a “small” root of unity automatically adjoins some […]

If number of conjugates of $x\in H$ in $G$ is $n$ then number of conjugates of $x$ in $H$ is $n$ or $n/2$

Let $G$ a finite group and $H$ subgroup of index $2$. Let $x\in H$ so that the number of conjugates of $x$ in $G$ is $n$. Show that the the number of conjugates of $x$ in $H$ is $n$ or $n/2$.

Possible orders of normal subgroups using only the elements in $G$ and their orders

I am told some information about a group $G$ of order $168$. All we are told about G is that: It has one element of order one, $21$ elements of order $2$, $56$ elements of order $3$, $42$ elements of order $4$ and $48$ elements of order $7$. and later it will be proved to […]

Simple module over matrix rings

I’m trying to prove that if $M$ is a simple module over $M_n(D)$ where $D$ is a division algebra, then $M\cong D^n$. I know that if $M$ is a simple module over $R$ then it is isomorphic to $Rv=\{rv|r\in R\}$ for all $v\in M$. I’m trying to find a $M_n(D)$ module isomorphism $f:M_n(D)v\rightarrow D^n$. I’ve […]