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

**MollyMATH** · May 25th 2009, 07:08 PM

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!

**NonCommAlg** · May 25th 2009, 11:21 PM

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 $\sqrt{5} \in \mathbb{Q}.$ then $[\mathbb{Q}(\sqrt{5}):\mathbb{Q}]=1.$ but, by Eisenstein's criterion, $p(x)=x^2-5$ is irreducible over $\mathbb{Q}$ and thus $[\mathbb{Q}(\sqrt{5}):\mathbb{Q}]=2.$ contradiction!

**ThePerfectHacker** · May 26th 2009, 09:09 AM

Originally Posted by NonCommAlg

this is really an overkill proof: suppose $\sqrt{5} \in \mathbb{Q}.$ then $[\mathbb{Q}(\sqrt{5}):\mathbb{Q}]=1.$ but, by Eisenstein's criterion, $p(x)=x^2-5$ is irreducible over $\mathbb{Q}$ and thus $[\mathbb{Q}(\sqrt{5}):\mathbb{Q}]=2.$ 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 $\sqrt{5}$ is irrational using the most overblown results from math that you can think of, or prove irrationality using a very long and confusing proof.

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