Let p and q be distinct primes. Prove that \sqrt{p/q} is a irrational number.
It isn't a homework. I just need to prove it!
Thank you,
Olcyr.
#3
csopi
81
2
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
olcyr
5
0
I din't understand why b isn't divisible by p.
Thank you for your answer!
#5
csopi
81
2
because if b is divisible by p, than GCD (a,b) is at least p, but we assumed that it equals to 1
It is well known that a vector space always admits an algebraic (Hamel) basis. This is a theorem that follows from Zorn's lemma based on the Axiom of Choice (AC).
Now consider any specific instance of vector space. Since the AC axiom may or may not be included in the underlying set theory, might there be examples of vector spaces in which an Hamel basis actually doesn't exist ?