Eisenstein Criterion to verify a that a polynomial is irreducible

Asked Apr 23 '20 at 17:32

Active Apr 23 '20 at 17:32

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Let $f(x)= x^4 -10x^2+1.$

Using the rational root theorem, I see that the polynomial has no rational root. Therefore, it does not have any linear factor. If it's reducible over $\mathbb{Z},$ then it would a product of two irreducible quadratic polynomials.

I did this pathetic arithmetic, and then I have shown that the polynomial is in fact, irreducible over $\mathbb{Z},$ ie irreducible over $\mathbb{Q}$ by the Gauss's lemma.

Is there any better way to check that the polynomial is irreducible over $\mathbb{Q}$? Can we use Eisenstein Criterion by translating the polynomial? Any help/suggestions would be appreciated. Thanks so much.

asked Apr 23 '20 at 17:32

Matha Mota

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This question already has many answers. See here – morrowmh Apr 23 '20 at 17:37
Thank you so much. I will close my post now. – Matha Mota Apr 23 '20 at 17:40

Eisenstein Criterion to verify a that a polynomial is irreducible

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