Search found 7 matches

by tziaxri
Thu Jun 09, 2016 10:25 pm
Forum: Algebraic Structures
Topic: Order of a finite division ring
Replies: 1
Views: 2347

Order of a finite division ring

Prove that the order of a finite division ring is power of a prime.
by tziaxri
Thu Jun 09, 2016 12:11 pm
Forum: Algebraic Structures
Topic: Invertible elements of a ring
Replies: 2
Views: 2835

Re: Invertible elements of a ring

Let \(\displaystyle{ 1-xy \in U(R)\Rightarrow \exists r \in U(R) : (1-xy)r=1=r(1-xy) ( \star\star )}\) .Relation \(\displaystyle{ (\star\star)}\) gives \(\displaystyle{ xyr=rxy (\star)}\). \(\displaystyle { (1-xy)r = 1 \Rightarrow y(1-xy)rx=yx.}\) .Observing that \(\displaystyle{ (1-yx)+yx = 1}\) an...
by tziaxri
Thu Jun 09, 2016 7:05 am
Forum: Algebraic Structures
Topic: Isomorphism and cyclic groups
Replies: 1
Views: 2116

Re: Isomorphism and cyclic groups

For the first part we have: Let the function \[ f : { \displaystyle{\left(\mathbb{Z}\times \mathbb{Z}\right)}}\longrightarrow \left(\mathbb{Z}{_2}\times \mathbb{Z}\right) \] defined as: \[ f((m,n)) = (\left[m]_{2},n-m\right) \,.\] It is easy to see that f is a well defined function. Also f is a grou...
by tziaxri
Tue Jan 19, 2016 4:33 pm
Forum: Number theory
Topic: Euler $\phi$ function
Replies: 1
Views: 3223

Euler $\phi$ function

Prove that

\[ \forall n\geq1 : \displaystyle\mathop{\sum}\limits_{d|n}\phi(d)=n \] where: \[ \phi(n) = \rvert\bigl\{k\in \mathbb{N} \rvert 1 \leq k \leq n \wedge (k,n)=1 \bigl\}\rvert \]

is the Euler phi function.
by tziaxri
Fri Jan 15, 2016 10:46 pm
Forum: Number theory
Topic: Infinitely many primes of the form
Replies: 1
Views: 2483

Infinitely many primes of the form

Prove that there are infinitely many primes of the form: \[3k + 2\].
by tziaxri
Fri Jan 15, 2016 10:37 pm
Forum: Number theory
Topic: Divisibility
Replies: 1
Views: 2522

Divisibility

Prove that there are no positive integers $a,b,n > 1$ such that: \[ (a^{n}-b^{n})\mid (a^{n}+b^{n})\]


P.S. I don't have a solution.
by tziaxri
Fri Jan 01, 2016 10:03 am
Forum: Real Analysis
Topic: Metric Function
Replies: 1
Views: 2307

Re: Metric Function

Let \(\displaystyle{H^{\infty}:=\left\{a=\left(a_{n}\right)_{n\in\mathbb{N}}: a_{n}\in\mathbb{R}\ \land \left|a_{n}\right|\leq 1\ \forall n\in\mathbb{N}\right\}}\) and the function \(\displaystyle{d\left(a_{n},b_{n}\right)=\sum_{n=1}^{\infty}\frac{\left|a_{n}-b_{n}\right|}{2^{n}}\,\,,\left(a_{n},b_{...