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Hello. I have this question from the book:

Prove that the polynomial [tex]x^4 + 2x^2 + 2[/tex] is irreducible in Q[x]. (Q being the rational fancy Q)

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So I used Eisenstein Criterion (because we are dealing with the rationals) and said that the coefficents are: 1 (for x^4), 2(for x^2), 2 with the last 2 coefficients being the important ones. So I then said that lets use 2 as the prime, and then 2 | 2 (the middle coefficient), and 4 (2^2) does not divide 2 (the last coefficient), so [tex]x^4 + 2x^2 + 2[/tex] is not reducible in Q[x].

That seems quite acceptable to me, but the book has no examples, did I do the problem correctly? Danke!

Prove that the polynomial [tex]x^4 + 2x^2 + 2[/tex] is irreducible in Q[x]. (Q being the rational fancy Q)

---

So I used Eisenstein Criterion (because we are dealing with the rationals) and said that the coefficents are: 1 (for x^4), 2(for x^2), 2 with the last 2 coefficients being the important ones. So I then said that lets use 2 as the prime, and then 2 | 2 (the middle coefficient), and 4 (2^2) does not divide 2 (the last coefficient), so [tex]x^4 + 2x^2 + 2[/tex] is not reducible in Q[x].

That seems quite acceptable to me, but the book has no examples, did I do the problem correctly? Danke!

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