## Factor a given polynomial into irreducibles over different rings

Completely factor and over , , and .

Note that over any ring with 1 we have and .

Over , is irreducible since is Eisenstein at 2. Also, is irreducible since is Eisenstein at 2. Thus is completely factored into irreducibles.

Similarly, is irreducible since is Eisenstein at 3, and is irreducible since is Eisenstein at 3. Thus is completely factored into irreducibles.

In this previous exercise, we computed the irreducible monic polynomials of degree at most 3 over and .

Over , has no linear factors since and . Suppose now that has a quadratic factor; say . Using the polynomial long division algorithm, we see that . Thus and . If , then the second equation gives , a contradiction. Thus , so that . Now we have , so that . Indeed, , and these factors are irreducible.

Evidently, and are completely factored over and and are completely factored over .

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## Comments

The 7th row of the proof has a typo. Should be x^2 + 3x + 3.

Thanks!