EINSTEIN'S IRREDUCIBILITY CRITERION OVER POLYNOMIALS.

Could I get a perfect explanation of einstein's irred. criterion. with its help can you tell whether the polynomial p(x)=x^2 + 2x +1,is irreducible or not.

Note by Kinjal Saxena
4 years ago

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I believe you are referring to Eisenstein's Criterion?

If so, you can use it to show that if \(p\) is a prime number, then the polynomial

\[ x^{p-1} + x^{p-2} + \ldots + x^2 + x + 1 \]

is irreducible over the rational numbers.

Calvin Lin Staff - 4 years ago

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Are you sure of your question? Your polynomial \[ p(x) = x^2 + 2x + 1 = (x + 1)^2 \] is clearly reducible!

On the other hand, \(q(x) = x^2 + 2x - 1\) is not, and we can use Eisenstein to show this, since \[ q(x+1) = (x+1)^2 + 2(x+1) - 1 = x^2+4x+2 \] is irreducible by Eisenstein. Since \(q(x+1)\) is irreducible, so is \( q(x)\).

A similar trick will handle Calvin's problem (think GPs).

Mark Hennings - 4 years ago

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http://yufeizhao.com/olympiad/intpoly.pdf

Theorem 2 on the link above is exactly what you want

Daniel Remo - 4 years ago

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