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

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    -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!
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    Quote Originally Posted by MollyMATH View Post

    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!
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    Quote Originally Posted by NonCommAlg View Post
    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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