Search found 7 matches
- Thu Jun 09, 2016 10:25 pm
- Forum: Algebraic Structures
- Topic: Order of a finite division ring
- Replies: 1
- Views: 3088
Order of a finite division ring
Prove that the order of a finite division ring is power of a prime.
- Thu Jun 09, 2016 12:11 pm
- Forum: Algebraic Structures
- Topic: Invertible elements of a ring
- Replies: 2
- Views: 4018
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...
- Thu Jun 09, 2016 7:05 am
- Forum: Algebraic Structures
- Topic: Isomorphism and cyclic groups
- Replies: 1
- Views: 2530
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...
- Tue Jan 19, 2016 4:33 pm
- Forum: Number theory
- Topic: Euler $\phi$ function
- Replies: 1
- Views: 4036
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.
\[ \forall n\geq1 : \displaystyle\mathop{\sum}\limits_{d|n}\phi(d)=n \] 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.
- Fri Jan 15, 2016 10:46 pm
- Forum: Number theory
- Topic: Infinitely many primes of the form
- Replies: 1
- Views: 3014
Infinitely many primes of the form
Prove that there are infinitely many primes of the form: \[3k + 2\].
- Fri Jan 15, 2016 10:37 pm
- Forum: Number theory
- Topic: Divisibility
- Replies: 1
- Views: 3174
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.
P.S. I don't have a solution.
- Fri Jan 01, 2016 10:03 am
- Forum: Real Analysis
- Topic: Metric Function
- Replies: 1
- Views: 2742
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_{...