Algebra Question: Does R/Z under Addition Have Infinite Elements of Order 4?

  • Thread starter Thread starter tgt
  • Start date Start date
  • Tags Tags
    Algebra
tgt
Messages
519
Reaction score
2

Homework Statement


Does R/Z under addition has an infinite number of elements of order 4? Where R denotes the real numbers and Z denotes the integers.






The Attempt at a Solution


Yes. Consider the cosets p/4+Z for primes p greater than 2. Since there are an infinite number of primes, there are an infinite number of such cosets. But the answer says no for this question.
 
Physics news on Phys.org
But each prime > 2 is 1 or 3 modulo 4. So if p is a prime, then the coset [p/4] = p/4 + Z is either equal to [1/4] or [3/4], isn't it?
 
Nice one.
 
Thanks. I'll leave it to you to prove that all other examples fail as well :smile:
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
View full post »

Similar threads

I Localizing single variable quotient polynomial ring at a prime ideal
Replies
6
Views
762
Show that ##G\simeq \mathbb{Z}/2p\mathbb{Z}##
Replies
2
Views
1K
How Do Vector Spaces of Linear Maps Differ from Standard Vector Spaces?
Replies
9
Views
2K
Deducing [Field of Algebraic Numbers : Q] is infinite
Replies
8
Views
2K
Some true or false Field extension theory questions
Replies
1
Views
2K
I Algebraic property of real numbers
Replies
30
Views
2K
Prove there are infinitely many primes using Mersenne Primes
Replies
1
Views
2K
True or False? Every infinite group has an element of infinite order.
Replies
10
Views
4K
How can relations on three-element set have seven elements?
Replies
24
Views
2K
Can Discrete Logarithms Be Added in Modular Arithmetic?
Replies
16
Views
2K

Hot Threads

Recent Insights

Back
Top