Monday, August 31, 2015

number theory - Prove that $2^{1/n}$ is irrational

Proof by contradiction, Assume $2^{1/n}$ is rational so:



$$2^{1/n} = \frac ab $$

where a,b have no common factors.



$$2 = \frac{a^n}{b^n}$$



$2$ divides LHS, therefore $2$ divides RHS
so $2$ divides $a^n$ or $2$ divides $b^n$ which implies $2$ divides $a$ or $2$ divides $b$.



Stuck on what to do next.

No comments:

Post a Comment

analysis - Injection, making bijection

I have injection $f \colon A \rightarrow B$ and I want to get bijection. Can I just resting codomain to $f(A)$? I know that every function i...