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.

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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Comments

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 - 7 years 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).

Mark Hennings - 7 years ago

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

Theorem 2 on the link above is exactly what you want

Daniel Remo - 7 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$