# Thread: -Prove square root of 5 is IRRational using {Abstract Algebra}

1. ## -Prove square root of 5 is IRRational using {Abstract Algebra}

Hey fellow mathletes, need help! Do not reply using anything but abstract!!
-Prove that the square root of 5 is irrational using abstract algebra; it is fairly urgent so anything will help, thanks!

2. Originally Posted by MollyMATH

Hey fellow mathletes, need help! Do not reply using anything but abstract!!

-Prove that the square root of 5 is irrational using abstract algebra;

it is fairly urgent so anything will help, thanks!
this is really an overkill proof: suppose $\displaystyle \sqrt{5} \in \mathbb{Q}.$ then $\displaystyle [\mathbb{Q}(\sqrt{5}):\mathbb{Q}]=1.$ but, by Eisenstein's criterion, $\displaystyle p(x)=x^2-5$ is irreducible over $\displaystyle \mathbb{Q}$ and thus $\displaystyle [\mathbb{Q}(\sqrt{5}):\mathbb{Q}]=2.$ contradiction!

3. Originally Posted by NonCommAlg
this is really an overkill proof: suppose $\displaystyle \sqrt{5} \in \mathbb{Q}.$ then $\displaystyle [\mathbb{Q}(\sqrt{5}):\mathbb{Q}]=1.$ but, by Eisenstein's criterion, $\displaystyle p(x)=x^2-5$ is irreducible over $\displaystyle \mathbb{Q}$ and thus $\displaystyle [\mathbb{Q}(\sqrt{5}):\mathbb{Q}]=2.$ contradiction!
Haha, that just gave me an idea. You know in math we try to find more and more elementary proofs. Well, how about we do the opposite for a change. Prove that $\displaystyle \sqrt{5}$ is irrational using the most overblown results from math that you can think of, or prove irrationality using a very long and confusing proof.