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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
3 years, 7 months 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.

Staff - 3 years, 7 months ago

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).

- 3 years, 7 months ago

Theorem 2 on the link above is exactly what you want

- 3 years, 7 months ago