1. Feb 7, 2010

### olcyr

Let p and q be distinct primes. Prove that $$\sqrt{p/q}$$ is a irrational number.

2. Feb 7, 2010

### olcyr

It isn't a homework. I just need to prove it!

Thank you,
Olcyr.

3. Feb 7, 2010

### csopi

It's quite easy. Assume, that $$\sqrt{p/q}=a/b$$, where a and b are relative primes, ie GCD (a,b)=1.

This is equivalent to $$pb^2=qa^2$$. Since p and q are distinct primes, p | a^2 => p | a => The right side is divisible by p^2, and this is a contradiction, because the left side is not (because b is not divisible by p, since GCD (a,b)=1)

4. Feb 7, 2010

### olcyr

I din't understand why b isn't divisible by p.

5. Feb 7, 2010

### csopi

because if b is divisible by p, than GCD (a,b) is at least p, but we assumed that it equals to 1

6. Feb 7, 2010

Thanks! :)