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 $y=4^nx+\frac{4^n-1}{3}$ 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 $\mathbb Z[x]$? $ 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 […]

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 […]

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.

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 […]

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 […]

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 $n/2$.

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 […]

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 […]

Intereting Posts

Cover $\mathbb{R}^3$ with skew lines
How did people get the inspiration for the sums of cubes formula?
How to find perpendicular vector to another vector?
Tail sum for expectation
On continuously uniquely geodesic space II
Proving $x\leq \tan(x)$
The trigonometric solution to the solvable DeMoivre quintic?
Calculating a spread of $m$ vectors in an $n$-dimensional space
Under what condition we can interchange order of a limit and a summation?
Non-inductive, not combinatorial proof of $\sum_{i \mathop = 0}^n \binom n i^2 = \binom {2 n} n$
An expression for $U_{h,0}$ given $U_{n,k}=\frac{c^n}{c^n-1}(U_{n-1,k+1})-\frac{1}{c^n-1}(U_{n-1,k})$
Prove that every nonzero prime ideal is maximal in $\mathbb{Z}$
Irrational roots of unity?
Prove $l_{ip}=\frac{\partial \bar x_i}{\partial x_p}=\frac{\partial x_p}{\partial \bar x_i}$
Why is free monoid called Free?