# Difference between revisions of "Ostrowski's criterion"

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Ostrowski's Criterion states that: | Ostrowski's Criterion states that: | ||

− | + | Let <math>f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\in \mathbb{Z}[x]</math>. If <math>a_0</math> is a prime and | |

<cmath>|a_0|>|a_n|+|a_{n-1}|+\cdots+|a_1|</cmath> | <cmath>|a_0|>|a_n|+|a_{n-1}|+\cdots+|a_1|</cmath> | ||

then <math>f(x)</math> is irreducible. | then <math>f(x)</math> is irreducible. |

## Latest revision as of 11:23, 15 June 2021

Ostrowski's Criterion states that:

Let . If is a prime and then is irreducible.

## Proof

Let be a root of . If , then a contradiction. Therefore, .

Suppose . Since , one of and is 1. WLOG, assume . Then, let be the leading coefficient of . If are the roots of , then . This is a contradiction, so is irreducible.

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