# Algebra question

1. Feb 10, 2009

### tgt

1. The problem statement, all variables and given/known data
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.

3. 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.

2. Feb 10, 2009

### CompuChip

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?

3. Feb 15, 2009

### tgt

Nice one.

4. Feb 15, 2009

### CompuChip

Thanks. I'll leave it to you to prove that all other examples fail as well