# Eisenstein's criterion

Let be integers. Then, **Eisenstein's Criterion** states that the polynomial
cannot be factored into the product of two non-constant polynomials if:

is a prime which divides each of

is not divisible by

is not divisible by

## Proof

Assume and for non-constant polynomials and . Since has only one factor of , we know that or . WLOG, assume . Then, we know that . This means . Similarily, we see, since , for all . This means that , so . However, we know that , a contradiction. Therefore, is irreducible.

## Extended Eisenstein's Criterion

Let be integers. Then, **Eisenstein's Criterion** states that the polynomial
has an irreducible factor of degree more than if:

is a prime which divides each of

is not divisible by

is not divisible by

### Proof

Let , where and . Since has only one factor of , we know that or . WLOG, assume . Then, we know that . This means . Similarily, we see, if , for all . This means that , so . However, we know that , a contradiction. Therefore, .

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